For live music, check out the Shibuya and Ebisu areas. There are lots of bars and clubs featuring local bands or DJs (depending on your tastes) as well as a good atmosphere.

For local culture, the afore-mentioned Yoyogi park, as well as Ueno park are excellent places to see people just being people. Also, in Ueno Park, you can pick up lots of street art as souveniers or presents for people back home.