In the USA:  (My apologies, the websites are mostly all in English, unlike most of those for the European museum websites.)

National Gallery of Art, Washington, D.C. -   http://www.nga.gov/exhibitions/index.shtm  This is a terrific website with a search feature, calendar, and - best of all, IMHO - a feature entitled "Past Exhibition Resources" which goes all the way back to 1941 and which enabled me to finally determine the source of some of my favorite paintings from an exhibit I lucked into in 1975 when it toured four locations in the USA, Master Paintings from the Hermitage and the State Museum, Leningrad [http://www.nga.gov/past/data/exh400.shtm].

In the general site, click on one of the Permanent Collection's categories, for instance "Henri Matisse, Cutouts" and get a description of the exhibit.  Then click on "Works by Matisse in the Gallery's Collection" to see the list, and click on any of those that are noted as "image available" to see a photo of the work (you can enlarge the thumbnails to a fairly decent size.)

Metropolitan Museum of Art, NYC  http://www.metmuseum.org/
This site also has a search feature that is very user-friendly.  Enter just the name of an artist, for instance "Jules Bastien-Lepage," and a list of his works appears (they have just the one, Joan of Arc, but it is one of the most breathtaking of the realistic paintings I've ever seen, and is quite large, about 254 x 280 cm.  Go here to see it: http://www.metmuseum.org/Works_of_Art/collection_database/Joan_of_Arc_Jules_Bastien_Lepage/ViewObjec t_enlarge.aspx?depNm=all&Title=Joan_of_Arc&pID=0&kWd=Jules+Bastien-Lepage&OID=110000 054&vW=1&Pg=1&St=0&StOd=1&vT=1&RID=1
My main problem with the Met is that they are open only business hours from Tuesday through Thursday and are closed on Monday, which makes it hard to visit if one is in NYC on business from Mon - Fri.

Contrast that with the Art Institute of Chicago   http://www.artic.edu/aic/index.php, which is open every day of the year except Thanksgiving, Christmas and New Year's Day!  And on Thursdays, they stay open until 8 pm, with free admission to the main gallery from 5pm to 8pm.  On the website, many images can be found (use the search feature) and many can be enlarged and COPIED!  Make your own calendars, coffee mugs, etc., with images such as a 604 x 758 px JPEG image of Madame Roulin by Van Gogh.  

My best experience in about 4 visits:  There was an exhibit of Mary Cassatt, very popular, I didn't think I'd be able to get in as I'd heard the lines were long and I was only in Chicago on business for the week.  After a day cloistered inside a microfilm room at the courthouse, I exited to a terrific thunderstorm and headed straight for the museum on the one night it stayed open until 8 pm.  Thanks to the fierce storm, I had Mary Cassatt almost to myself!  I was holding back tears in front of one of the paintings and an old guard (who sort of reminded me of Lou Gossett) came over and said "don't be embarrassed, these made me cry, too."  
On another occasion I saw an exhibit of tiaras (yes, tiaras) that was gorgeous, especially some of the modern ones by Vivien Westwood, which were also shown at the V&A in London see the web page here

Museum of Modern Art, NYC     http://www.moma.org/    It was here that I saw the Chuck Close exhibit that was truly awesome (sure, size matters.)  And also saw Van Gogh's Starry Night and Picasso's Guernica.  I wasn't prepared for the latter, didn't even know it would be there, and it stabbed me in the heart the moment I came upon it.  
The web site is another excellent one that allows searching, printing and image-saving of many works.  The Chuck Close images on the page that detailed the exhibit had only a few of the works and the images couldn't be enlarged (boo hoo!)

Guggenheim Museum, NYC    http://www.guggenheim.org/new_york_index.shtml
Okay, I have a confession.  Despite the large amount of time I've spent in NYC as a tourist and on business, I've been only once to the Guggenheim, and that was for the "Art of the Motorcycle" exhibit with my machine-crazy German husband.  I've never seen such a ... "different" bunch of people at an art museum before, and it was one of their most-visited shows at the Guggenheim.  Groups would be gathered around each motorcycle, crouched down, pointing their tattoed arms at the machines (which added greatly to the art-mosphere, IMO.)  I loved it, especially because I was able to make it happen for my sweetheart.  Their website isn't as much fun as many of those above, but it's serviceable.

Cooper-Hewitt, National Design Museum, NYC    http://www.cooperhewitt.org/
Go for the house itself, the old Andrew Carnegie mansion on Fifth Avenue.  And go on a Tuesday night if you can, as I did, when it's free.  Whoops!  Sorry, perusing the website, looks like they no longer open past 5pm on Mon-Thurs, and no "free" night is mentioned.  It's a small museum, and I'd put it AFTER the others above if I had limited time in NYC, but it's still a lovely place with changing exhibits.

Los Angeles County Museum of Art  http://www.lacma.org/
They also have a good permanent collection and a user-friendly website with the opportunity to save the images on one's own computer (my favorite feature on many sites.)  I think this is where I saw a Jackson Pollack exhibit long ago, but there's an example of my memory failure and they don't have a long-ago exhibits search feature like the National Gallery's.

The Museum of Fine Arts, Houston  http://www.mfah.org/home.asp?par1=1&par2=1&par3=1&par4=1&par5=1&par6=1&par7= &lgc=0&eid=&currentPage=
This website does those 360-degree panoramas that I love, though these serve mainly to show that the museum has more space than paintings.  They brag at being 5th largest in the USA in terms of space, but the collection can't begin to compare with those in Chicago, New York, Washington, D.C. and such.  As I now live in art-museum-starved Austin, though, I'm quite thankful for Houston's offerings and hereby officially apologize for the previous disparaging observation.

The Menil Collection, Houston http://www.menil.org/home.html    I love this museum primarily for its collection of Rene Magritte, the largest there is of his works, I believe.  

In Europe:

Van Gogh Museum, Amsterdam    http://www3.vangoghmuseum.nl/vgm/index.jsp?lang=en
It's impossible not to love this museum if you're a Van Gogh fan.  The first time I was in Amsterdam, I visited the museum twice, first as my normal self, then slightly "altered."  It was wonderful both ways.  If you can go only once, I can't tell you which way would be better, as that's a very personal and subjective thing.  
The website is another one of those terrific ones that lets you enlarge and save images.  You can also choose to view the site in a number of languages.

Stedelijk Museum of Modern Art, Amsterdam      http://www.stedelijk.nl/   This website is terrific, even allowing one to take YouTube tours.  They also have an archive that provides info about previous exhibitions back to 1996.  

Rijksmuseum, Amsterdam     http://www.rijksmuseum.nl/index.jsp
Another confession here: the doll's house of Petronella Oortman is one of my favorite things in this museum, much as I love the art collection, too.  The website is user friendly, in a choice of languages, and offers some audio tour samples as well as interactive panoramas.  

Uffizi Gallery, Florence    http://www.polomuseale.firenze.it/musei/uffizi/default.asp
Okay, personal affliction, I just don't enjoy most "religious" art, though I was wild about Botticelli's Birth of Venus, and I took many photos of the ceilings, which were gorgeous.  The website is in Italian and English only.  I couldn't find a way to actually view the particular works, though information about locations of the works in the museum and of the dates of the works is provided.

Galleria dell'Accademia, Florence  (David)  http://www.polomuseale.firenze.it/musei/accademia/default.asp  This website seems to be operated the same way as the Uffizi Gallery website.  And, of course, if you want to see David, this is where you go.  

Alte Pinakothek, Munich    http://www.pinakothek.de/alte-pinakothek/  
Neue Pinakothek, Munich    http://www.pinakothek.de/neue-pinakothek/index_en.php?  
The websites are available in German and English.  They aren't really interactive and you can't visit the works online.  But if you're in Munich, they're worth a visit.  But I prefer the following museum in Munich:

Bavarian National Museum, Munich    Bavarian National Museum  Website in German and English.  I adore museums that display the practical arts and artifacts of life.  This one pleases!  There are musical instruments, porcelains, sculptures, a collection of armor, furniture and pottery.  I recall some very large tables with models of German cities from the Middle Ages, but maybe I'm recalling a different museum.  Alas, I couldn't find a search function.

Kriminalmuseum, Rothenberg-ob-der-Tauber, Germany   http://www.kriminalmuseum.rothenburg.de/   The website, in German, English and (Japanese?) has no search function and is pretty scant, but the museum itself is fascinating, if a bit grotesque.  Being a Baker's wife and a bit of a "Scold", I was particularly amused by the punishments meted out to bakers whose wares were too much under (or over!) weight, as well as the iron masks put on women who lashed out too readily with their tongues.  If you visit this beautiful, if touristy, little medieval village, see this museum and then go to the following to get the bad images out of your head:

Doll and Toy Museum, Rothenberg ob der Tauber  http://europeforvisitors.com/germany/rothenburg/rothenburg-doll-and-toy-museum.htm   This museum is small but delightful, especially if, like me, you are unduly fond of little dollhouses.  The website isn't the museum's, just a visitors' guide recommendation site.

Musee des Arts Decoratifs and  Musee Alsacien, Strasbourg   http://www.musees-strasbourg.org/F/intro.html   This website covers the museums of Strabourg, but these two are the ones I visited and enjoyed, particularly the latter, which was highly educational and fascinating with regard to life through the ages in the Alsace region.  The museum is in an old building of several stories and takes a while to walk through, which is a plus for me.  One sees the implements and artifacts of daily life, wine and beer making, the chandlers' trade, many aspects of life and work.  Just go.  The website appears to be in French only, though there's an upcoming events program that can be accessed in English in a pdf.

I would feel like an idiot giving "information" about the museums in Paris and London to a group of Europeans, so I'll provide any personal highlights regarding the following, and a note about the websites, but am really more interested in learning what truly knowledgeable people have to share about these places:

Centre Georges Pompidou, Paris http://www.centrepompidou.fr/Pompidou/Accueil.nsf/tunnel?OpenForm  (in French, English and Spanish)  This site is great, lets you search for and see works online (but no commercial use, please).  I had a truly fun time at this museum, once with my daughter and another time with my best girlfriend, walking through and being scornful of the modern art (some was really good, but a dog bed sitting on the floor in a corner?  Puh-lease!  Okay, I apologize to anyone who enjoyed seeing a dog bed out of its ordinary context and considered it to be art.  It must be some failing in me that I just don't get it.  The building is something to see, for sure. Oh, and my favorite things I ever saw there were some sculptures by Testsumi Kudo.

Louvre Museum, Paris     http://www.louvre.fr/llv/commun/home.jsp   Another good website with self-conducted online tours available.  French and English.  The museum itself?  TOOOO MUCH!!  Save this museum for last so you don't suffer museum fatigue and then fail to visit the other wonderful Paris museums.  Yes, it's wonderful, but overload is a serious problem.

Musee d'Orsay, Paris     http://www.musee-orsay.fr/en/home.html (French, German, Spanish and Italian).       My favorite.  The Impressionists, the ability to take non-flash photos, the collections of furniture, the cutaway model of the Opera - it's just wonderful.  The floor plan confuses me, but that's also a good way to see things you think won't interest you as much as others, but then they do.  The views from the terraces are great, too.

Musee Rodin, Paris    http://www.musee-rodin.fr/welcome.htm  The website isn't as manageable as some others for me, but there's a calendar of upcoming events that's helpful.  I don't find a choice of languages, but things seem to be intermingled French and English.  "Maybe I'm doing it wrong," as Randy Newman sang.

London:  (Again, who am I to comment on this stuff?)

British Library   http://www.bl.uk/  Great online resources, very searchible, more oriented to actual research, I think..  Exhibits are brainy, IMHO, geared more for adults.  It's also peaceful and quiet and pretty and the little café is nice, as I find to be generally true in most of the British museums.

British Museum    http://www.britishmuseum.org/default.aspx  Website can be accessed in German, French, Spanish and several others.  I personally make sure to hit the V&A, Tate, Portrait Gallery, and Imperial War Museum first.

National Gallery     http://www.nationalgallery.org.uk/  Good website in 6 languages, upcoming events, how to plan your visit, etc.  I'm very sorry to be missing a current exhibit of art that's come out of primary schools!  It closes July 13, 2008.  Lots of Important paintings.  I could be wrong about the Dutch realists I loved being at the Tate Gallery, they may have been here.

Tate Gallery     http://www.tate.org.uk/  Excellent website with searchable features, calendar, covers a number of Tate museums.  It also can be accessed in plenty of languages  Lots of wonderful art in the Museum, some of my favorite Dutch realists.  And to cater once again to my stomach, the cafeteria is terrific, always has some yummy vegetarian soup choices.  Lots of clever and educational gifts for the little ones in the gift shop.

Victoria and Albert Museum       http://www.vam.ac.uk/  I hardly know where to start, this museum (and others now part of it) have provided me with so many thrills.  The website is loaded and comes in plenty of languages.  I particularly love the collection of glass and porcelain through the ages, the clothing exhibits, the Frank Lloyd Wright furniture - I know I'm leaving out a lot of things, but it's there for you to find your favorite parts and the website can help.  

Barbican Centre         http://www.barbican.org.uk/  Okay, not really a museum so much, lots of activities in the performing arts and film categories and delightful.  You can keep up with the schedule on the web site, which is wonderfully informative, but of course the events favor English speakers, as does the website.

Cabaret Mechanical Theatre       http://www.cabaret.co.uk/  This is precious, just for fun, located at Covent Garden (last I knew.)  The website is fun, too, let's you take an online tour of some of the little devices which are part of the exhibit.  Oh my goodness!  The Mechanical Theatre has greatly expanded since my last visit, and seems to be opening up and touring EVERYWHERE.  Click on "Exhibitions" to see.  It's espcially great for children.

Dulwich Picture Gallery    http://www.dulwichpicturegallery.org.uk/  I include this small gallery because of my best (male) friend Andy Archibald, the poet, who was a professor of English (and director of the plays) there for a good while, who gave us a personal tour.  But don't miss any of the other museums and such mentioned in this diary in favor of this gallery - my personal advice.

Imperial War Museum    http://www.iwm.org.uk/  I learned more about the Spanish Civil War at this museum than I ever learned in school, in a special exhibit that had a big re-creation of Picasso's Guernica.  The special exhibits are painstakingly done, and if you are avoiding the place because you think it might celebrate war, I would advise that it seems to be more anti-war.  One drawback; I couldn't find a way to access the site in other than English.  But the website is comprehisive and notes the upcoming exhibitions.

Museum of London     http://www.museumoflondon.org.uk/English/

National Portrait Gallery      http://www.npg.org.uk/live/index.asp  I don't know where to start with how much I love this place.  Once saw an exhibit on the work being done to give new faces to disfigured people - a bit graphic, but moving and uplifting.  I simply love the portrait of Steven Hawking, which reminds me of the tone of a self-portrait done by my Dad.

I think the portrait gallery may be re-opening in a new location in December 2008, so check before you go.

Tate Modern    http://www.tate.org.uk/modern/  Lots of languages for this website, which is also informative and has a good calendar.  Lots to love in the museum, and plenty at which to scoff, if you're a "limited" fan of the very modern art scene, which is often the case with me, I confess.  But scoffing can be fun, too, especially if you're doing it with friens.  The "Street Art" thing they have going now looks wonderfully intriguing, wish I were there.

Bethnal Green Museum of Childhood  (I get the feeling it's now part of V&A)   http://www.vam.ac.uk/moc/your_visit/index.html
http://www.qype.co.uk/place/72098-Bethnal-Green-Museum-of-Childhood-London
I love this place, especially for the toys and doll houses, musical instruments through the ages.  It's a bit out of the way but worth the trip.  The sites I included give reviews, but I've discovered that you should go to the V&A website listed with that museum above, which includes this on its site.

So... that's all I have time for.  I hope y'all will give me some good suggestions for must-see things (not just museums!) in the places I mentioned at the top, and thanks ahead of time.

Karen in Austin  (almost midnight, off to bed.  Thanks to In Wales for inspiration and encouragement)