here some place i find interesting (those marked i have not visited yet):
watari-um art museum (*) http://www.watarium.co.jp/museumcontents.html
museum of modern art (*) http://www.momat.go.jp
ma gallery: b-e-a-u-t-i-f-u-l-l architecture gallery. simply a must. http://www.toto.co.jp/gallerma/
prismic gallery: close to gallery ma, much smaller but interesting to get to know younger architects. the web site has links to the studios who exhibited, take a look. http://www.prismic.co.jp/gallery/
kandada(*): artist collective exhibition space, suggested by tomoko, seems very interesting. http://www.commandn.net/
no.12 gallery (*): small gallery, also possible to rent it.
in case you come to tokyo and you need to buy a lot of things for little money and therefore impress your friends and relatives here are the magic names you have to look for:
100yen shop: they are spread everywhere, it is similar to 50 cents shop but cheap japanese goods are much more interesting
donki: http://www.donki.com you can buy almost anything and it is also interesting to experience high-density shop layout especially after aoyama high fashion boutique
tokyu hands: another big department store
GO. for free lesson go to ben cafè (www.bencafe.com) Sunday 11:00 am.