^Honestly when it comes to spectacle etc, Shibuya is a bit more exciting than Shinjuku (more video screens and teenagers)
Miyashita Park there is also rather lovely, and interesting cuz it's located on an overpass...
This is a given, but be sure to check out Akihabara and Nakano Broadway for idol stuff.
Basement of Ikebukuro Sunshine City has lots of very small shops (practically stalls) with great clothes for quite cheap (upstairs is more obvious mall chains.) Koenji district, too, has sooo many great clothing shops. Then obviously Harajuku. Also Shimokitazawa (which has nice cafes as well)...
If you're into other types of music, tokyogigguide.com is a great resource for finding cool shows (and by extension, interesting bars/clubs). I went to an awesome noise and experimental electronic show at this place called Super Deluxe in Roppongi the other week on a whim and met a lot of really fun/nice/cool musicians.
I like the National Film Center in Ginza a lot, shows mostly older and/or notable Japanese films daily, on 35mm, for very cheap!
For touristy stuff, the small streets around Ueno station are a sight. Also, the huge plaza at Ebisu is crazy at night, looks like a German fairy tale somehow...
There's a pretty cool contemporary art museum in Aoyama called Watari-um, has an ok exhibition going on right now of that guy JR--worth checking out imo
Lots of free galleries in the basements or top floors of clothing stores around Omotesando but they usually kind of suck.
The National Art Museum in Osaka rules hard, if you wind up over in that direction.
If you go to Nagoya, check out Osu (Nagoya's version of Nakano Broadway meets Akihabara, kinda--open air electronics and other weird and not weird stuff-centric shopping arcade). If you go on Wednesday night, you can also see OSU, the (awesome) local idol group, perform (for free) in the central plaza. (And if you actually do this and she happens to be performing that night, please give a few cheers to Tomomi for me~)
My biggest advice would be to walk everywhere, even if it will take extra time and make you get lost. This is of course the best way to experience any city, but especially one that is not structured on a grid system and has very few street names.