Get thee a MetroCard. The subway is the easiest/cheapest way to get around the city. $29 gets you seven days of unlimited travel on trains and local buses. Buy one at any subway booth.
For the full tourist experience, you might want to try the CityPass. $89 inc Empire State Building, Natural History Museum, Guggenheim, MoMA, The Met & Circle Line Cruise. If you decide to get one, choose the Circle Line Cruise option instead of Statue of Liberty since you see it on the cruise anyway, and you'll see more of Manhattan from the water. Statue of Liberty itself is cool but the specialised tours are very busy and not worth the wait, I don’t think.
Seinfeld fans can hit up Tom's Restaurant (2880 Broadway). If coming from Brooklyn or Downtown, you'll want to take the subway to Cathedral Pkwy station, then walk North. The inside actually looks nothing like you think, but go in for the pancakes anyway.
|American Museum of Natural History|
|The Temple of Dendur|
|View of the Empire State Building from the Top of the Rock|
|Grand Central Station|
|The High Line Park|
If hunger strikes, go to Quantum Leap (226 Thompson St) for awesome, well-priced vego food. There's also Caffe Reggio (119 MacDougal St) if you'd rather a coffee and panini. Or walk a little further West for beer and jazz at the Fat Cat (75 Christopher St).
|See the Statue of Liberty from the Staten Island Ferry|
|The Brooklyn Bridge|
|Brooklyn Waterfront Park|
Need coffee? The best is at Oslo Coffee (328 Bedford Ave).
|Eggs Rothko at egg cafe|
|The Meatball Shop Menu|
Finally, the cocktail hour! I really liked Maracuja Bar (279 Grand St). The staff were friendly, and willing to recommend various beers to try. Make new friends at the pool table. Or hang out in the back courtyard with all the fairy lights.
Make a special trip to Park Slope for dinner at Bonnie's Grill (278 5th Avenue). They have the best burgers in NYC. No, seriously.