Live music
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The Tokyo Gig Guide lists the vast majority of intersting shows happening in Tokyo (in English!).
http://www.tokyogigguide.com/
If you're going to be here on the 19th some friends of mine put on a free monthly noise and etc. show here:
http://www.cinemabokan.com/. Attendance is highly encouraged.
Otomo Yoshihide currently has a really cool sound installation in Harajuku.
http://www.vimeo.com/1802336. This is also highly recommended. I think it will be there until the end of August. Only 500 yen entry fee. Map here:
http://www.n0idea.com/vacant/vacant1f.html
Super Deluxe in Roppongi hosts lots of cool shows
http://www.super-deluxe.com/
If you like live jazz these places have my highest recommendations:
http://www.pit-inn.com/sche_e.html
http://www.aketa.org/index.html
http://members.at.infoseek.co.jp/jazz_candy/ (in Chiba but worth the trip)
If you give me specific dates you'll be here I might be able to clue you into some other cool events.
Record Stores
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Modern Music (PSF Store) -
http://www.psfmm.com/
Warszawa -
http://www.warszawa.jp/ Has a nice cafe attached to it.
Otherwise I recommend wandering around Shimokitazawa, Koenji, Kichijoji and Shinjuku. The Disk Unions in Shinjuku (there are many!) and Kichijoji are particularly cool. Shimokitazawa and Koenji have a lot of nice mom n' pop style shops.
Other Stuff
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If you're into weird outdated second hand electronic whatnots Akihabara is great. Steer clear of the big shops and adventure into dark and dank rundown shops to find the good stuff.
Koenji is good for second hand shops and also has a lot of good restaurants, cafes and live houses. I think Koenji is my favorite neighborhood in Tokyo.
Shimokitazawa is kind of the same as Koenji, but a little too hip for my tastes. Definitely worth a visit though.
Ochanomizu and Jimbocho have a lot of cool used book and instrument stores.
Cultural Weirdness
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Walk around these places and you will find plenty of it.
Shinjuku - Go out the east exit of the station and walk past the giant TV. This area is known Kabuki-Cho. It is the Mecca of Japanese douche-bags. Just amazing. If you turn right out of the east exit (Kabuki-Cho exit) and walk along the big street (Shinjuku Dori) you will end up in Shinjuku 2-Chome, Tokyo's only gay neighborhood. Worth a look.
Yoyogi Park (short walk from harajuku station) is one of the best places to see Tokyo's weirdness on parade. It's really quite incredible and impossible to describe. Every mentally ill person in the city gathers here to do whatever crazy shit it is that they do. If you go on Sunday you can see the infamous Yoyogi greasers.
http://www.youtube.com/watch?v=kkxm1...eature=related
I think Yoyogi Park gets my highest recommendation for things you have to see in Tokyo.
Shibuya, Akihabara, Harajuku and Ueno are also great places to wander about if you want to see weird Tokyo.