See? This is why I vanity search.
(unless noted, the station names are the same as the neighborhoods)
Shibuya has lots of shops of all varieties (though mainly youth-oriented) and is good for people-watching.
Omote-sando has lots of small boutiques. Personally, I think you’re insane if you go to Tokyo to shop for Western-style clothes, but if you’re interested in some of the local designers you’ll probably find their stuff around here.
Kamiya-cho has Tokyo Tower, a nice Russian restaurant and not much else.
Asakusa is one of the older districts and has the huge Kaminari temple. It’s a big tourist spot, so there are lots of souvenir stands. There’s also a dock nearby where the Sumida River tourboats leave (the tours end around Hamamatsu-cho station at the mouth of the river).
Kudan (Kudan-shita station) has Yasukuni shrine, which has lots of WWII memorials, the remains of a few war criminals, and is where the Prime Minister goes when he wants to piss off Korea, China and the domestic anti-war groups.
Kudan-shita, Nijubashi-mae, Sakurada-mon and probably a few other stations are next to entrances to the Imperial Palace Grounds, which has a couple of art museums, a nice garden, and a moat filled with swans and turtles.
Akihabara is good for shopping for electronics.
Harajuku and Yoyogi are next to Yoyogi park, which on the weekends sometimes has dozens of small rock bands performing in it. Watching a half-dozen kids in full punk regalia all doing the Twist in perfect synch is a sight not soon forgotten.
The west side of Shinjuku has a number of skyscrapers (including the Tokyo Metropolitan Government Building) with observation decks open to the public. In the winter, the air is usually dry and clear enough to give a great view. On the east side, there’s… well, I don’t think I should go into much detail about the east side of Shinkjuku. After all, I’m a priest. Or something.