Edo-Tokyo Museum is a great place to start if you're looking to know all about Tokyo's history.
Also of interest are the Tokyo National Museum (Japan's oldest museum), the National Science Museum, Tokyo, the National Museum of Emerging Science and Innovation (Miraikan), and the Science Museum.
The Japanese traditional performing arts are Kabuki, Bunraku, Noh and Kyogen. The weird and wonderful Noh plays combine highly stylised vocalising with impressive traditional costumes. Kabuki is a synthesis of dance and singing that was created in the 17th century, and Bunraku is a traditional puppet drama. Kyogen evolved in parallel with the Noh, but is light-hearted and comical. The National Theatre, National Noh Theatre, Meijiza Theater and National Engei Hall all produce performances of these traditional genres. If you’re more into classical concerts, opera or ballet, then you should check out the programmes at Tokyo Opera City, Bunkamura, or Suntory Hall.