Hey There...well i've lived in NYC for 14 years so i wanna help you out ;-)

Subways: The Subways are very easy. They al either run WEST/EAST or NORTH/SOUTh. The majoiry being the N/S ones. If you're going to be there a while get a MetroCard. Its yellow card that can be refilled easily at machine kiosks. Right now the subway is $2.75 per ride....u actually get free rides if you buy ticket packages in advance at the machine. yes u can buy a 1 week pass. THere are multiple subway lines but they are all color coded. Get yourself an APP for the nyc subway. That way you dont look like a tourist constantly staring at the maps in the subway. It'll also help u plan ur trip. Look for an app that has a ROUTE PLANNER.

Places to go other than what you said: The Cloisters...beautiful scenery (uptown), You should do uptown to columbia for sure....it just has a different flavor there. My favorite area is Greenwich village. Take the 1 train to Christopher street and walk around....Great places to eat. Hippy and cool vibes....same with the Easy village if u wanna see "hipsters" be able to get great coffee, sushi and ramen. Go to UNION SQUARE too....fun area, has a farmers market.

Iconic Food: Go to Mamouns for falafel in the village. Johns pizza (village). Carmines for Italian (but touristy...good food tho)...theres on in Times Square and one downtown.
I HATE TIME SQUARE but all natives do....but u should do it. Go to the M&M store and get the variety of candy ;)

No fav hotels bc i live there but you can ask backpacking sites i'm sure.

Despite the reuptation...MOST New Yorkers are cool...so if u get lost its ok to ask someone...make sure they dont look crazy tho ;-)

If you like weird and unique this is a good article: http://julep.triplemint.com/17-secret-places-in-nyc-you-didnt-know-about-but-should/

Best of Luck,
Sherisse