I think FAO Schwartz closed…
But anyway. I live in NYC, and my favorite places are:
-Chinatown/Little Italy-- amazing bargains, foods, and just really cool to visit particularly if your city doesn't have a Chinatown. You can get plenty of anime here, too, but check first to make sure it has an English sub track at all.
-Brooklyn Botanical Garden & Prospect Park (they border one another). More to chill out than for touristy reasons, but as someone else mentioned they have a really beautiful Japanese garden. Plus, the subway goes over the Brooklyn Bridge, which is neat.
-MoMA (Museum of Modern Art) or the Guggenheim-- the two flagship modern art museums-- both have great style. The Guggenheim is a giant spiral of exhibits, you can choose either up or down. MoMA is also a neat space and is enormous.
-Shake Shack in Madison Square Park (NOT the garden) has truly truly excellent burgers and thick milkshakes called "concretes". Best burger in the city, and it's hella cheap too. People line up for 45 minutes for a burger at rush times-- at noon you can be in & out in 15 minutes, but by 12:15 forget it! Also a fun place to spot ridiculous little yappy New York dogs and mock them.
-New York Botanical Garden in the Bronx is gorgeous and much bigger than the one in Brooklyn, but it's a bit off the beaten path.
-Grand Central Station is one of the last great remaining train stations. Stop in to see the grand concourse and its ceiling with the zodiac. Menchanko Tei at 45th and Lex (northeast corner) is three blocks away and is a Ramen chain from Tokyo, good place to grab lunch (although better when it's colder weather).
-Five Points in Queens is a roof where graffiti artists have gone absolutely nuts and created an incredible sight.
-Soho is the best place to spot trendy people and go hopping between little private art galleries, slightly off-kilter shops, and see the NYC gay scene. DO NOT GO if you're homophobic. (In fact, if you're homophobic, stay the hell out of my city... anyway.)
-Central Park, of course.
-Midtown Comics is worth a stop if you're planning on hitting the Empire State Building and you like comics, since it's right across the street. Huge selection of tpbs and nice people.
-Washington Square Park is a great community park that's famous for its chess hustlers. DO NOT get sucked in-- but it's fun to watch. There's also a big fountain area where street performers do magic or acrobatics shows on afternoons, and just in general throughout the park you see pretty good musicians and street performers. I like to hang out by the Bocce court and watch the old Italian guys. It's also near Cones, which might just be the best Ice Cream in the city, as well as Beard Papa, which is a Japanese cream puff store. OMFG delicious. At the Astor Place subway stop four blocks away there's a big black cube sculpture that you can push and it swivels, as well as St. Mark's Place, which is a great scene particularly at night. Yakitori Taisho will give you any part of a chicken on a stick until 3AM, and there are nice 2nd-hand record shops and such. (I spend a lot of time in this area because it's near NYU, where I go to grad school.)
OK, that's more than enough. Have a fun trip!
Avoid if at all possible:
-Times Square (what a headache!)
-Statue of Liberty (the line is ungodly)
-Midtown shopping (unless you like staring at ugly, boring $400 dresses)