Ah, fresh meat
Waaagh_Gonads has a lot of good tips.
Re: Game Shops - there are really only 2 game shops. There's the
GW in Greenwich Village on 8th
St. and Broadway (roughly) and the Compleat Strategist on 33rd
St. and 5th Ave. If you're interested in comics, there's Midtown Comics near the famous Times Sq. that Waaagh suggested as well as Jim Hanley's Universe on 33rd
St and 5th Ave right across the street from the Empire State Building.
RE: Punk clothing -
St. Mark's has a lot of punk-wear available. (East 8th Street, between 3rd and 2nd Avenues).
RE: Things to see -
I think they've re-opened the Statue of Liberty to tourists - but if you want to go up, make sure to reserve a ticket in advance. You can just pick it up near the ferry to Liberty Island.
Little Italy and Chinatown are good places for food and Chinatown is great for random cheap crap.
St. Mark's in the Lower East Side is also a great place for random food and drinks.
The various museums around Manhattan are quite great, though a bit on the pricey side for admission (I think).
See if you can catch a show somewhere in the city - afterall, theatre is a huge part of the city.
While it's a bit out of the way, I recommend a walk across the Brooklyn Bridge - it really is a nice walk, and gives you a beautiful view of the city. Plus, if you walk over to Brooklyn, you can stop by Grimaldi's Pizza - easily one of the best and most renowned of the NYC pizzerias (a bit pricey, but very very worthwhile).
Arthur Avenue is another great Italian neighborhood to visit, if you're interested in fine Italian fare and hand-rolled cigars.
If you don't mind the long-ish ride, Coney Island is definitely worth checking out (and ride the Cyclone if you're into roller-coasters). Eat at the famous Nathan's Hot Dog!
Lots of other things/places, depending on what you're into.
Re: where to stay away from -
Basically, avoid the bigger parks at night (noted exceptions are the Madison Square Park and Union Square Park). Probably want to avoid most of Manhattan above 120th Street (to be fair, it's not that bad, but there's also not much to see there, unless you're keen on going to the Apollo Theater in Harlem). During the day, it's not too bad anywhere in the city, really, so long as you don't walk alone into some strange, quiet back alleys. At night, I'd stick to more touristy areas.
Wear good shoes. You'll be walking A LOT.