Tokyo must be one of the best cities in the world to be an art lover. There's an enormous diversity of museums spread throughout the city, from expansive museums to tiny hole-in-the-wall galleries, so on any given day you'll be able to find something to suit your taste, budget and schedule.
A good place to start is Ueno, at the north-eastern apex of the Yamanote loop line. Around the edges of its pleasant park you'll find Tokyo National Museum, the Ueno Royal Museum, Tokyo Metropolitan Art Museum, and the National Museum of Western Art, all within easy walking distance of each other, and there are also occasional exhibitions of students' works at the nearby Tokyo University of Fine Arts.
There are naturally a number of places to take in traditional Japanese artworks. Close to Tokyo Station there's Mitsubishi Ichigokan, with the Idemitsu Museum of Art just a little further away. You can see beautiful old woodblock prints at the Ota Memorial Museum in Harajuku; Nihonga (classical Japanese painting) at the Yamatane Museum in Hiroo, or contemporary Nihonga at Naka-Meguro's Sato Sakura Museum; calligraphy and ceramics at the Gotoh Museum, and art and implements for tea ceremony at the Hatakeyama Museum; and works from the early 20th Century at the Yayoi and Takehisa museums, and at Meguro's Metropolitan Teien Art Museum, housed in a former Imperial residence.
Fans of Western and contemporary art are equally well served. Roppongi has the National Art Center and the Mori Art Museum, near Tokyo station you can go north-west to the Tokyo Museum of Modern Art or south-east to the Bridgestone Museum of Art, while a little further from the center is the Museum of Contemporary Art. There are also smaller, more specialized places, such as the charming Musee Hamaguchi Yozo, dedicated to mezzotint prints.
Most of the larger museums charge entrance fees of up to ¥1500 for their permanent collections and for special exhibitions, but there are plenty of places to see art for free. Kyobashi and Ginza abound with small independent galleries and tiny showcase galleries in shops and offices, such as the Lixil Gallery and Shiseido Gallery. In Meguro there's the Museum of Contemporary Sculpture, while a little way north of Akihabara station you'll find the quirky 3331 Arts Chiyoda, which holds contemporary art exhibitions in a renovated former junior high school.
One of the best ways to find out what's going on is to use Tokyo Art Beat. They publish monthly Art Maps which you can find at some bookstores, music stores and museums around the city, and their fully bilingual website can be searched by district or genre so that you'll be able to find something, somewhere of interest. Happy exploring!