Check out some of the other answers to similar questions here. I like Brooklyn, east village, Gramercy/union sq and Chelsea.
Brooklyn (brooklyn Heights, Carrol Gardens) is lovely- with great restuarants, nice leafy streets and brownstones, and if staying in the Heights, super one stop to Manhattan(court street/borough Hall stations)
With the Manhattan neighborhoods it's easy to walk to everywhere and subway stops everywhere. I sublet on the Upper East side off 86th& 3rd and that was also easy to get about. Anything on the 4,5,6 line is easy in my view.
Check out Murray Hill & Hells Kitchen ( by 9th/10th) over 7 years ago