I've spent most of my adult life in NYC. As I''m sure you're aware NYC is vast and offers an infinite variety of entertainment, site-seeing, and experiences. So it would help if you were more specific about what you're into or looking to see/do.
Notwithstanding the above caveats, when tourists think about venturing beyond Manhattan, they usually think Brooklyn, which could be the U.S.'s 4th largest city if it were stand-alone. And when tourists think Brooklyn, they usually think about the following:
Williamsburg - Ground zero for the East Coast hipster aesthetic, now populated with lots of creatives, professionals and families. Visually gritty and urban, lots of converted warehouses, new condo bldgs perptually under construction. Perhaps NYC's best culinary and bar hopping scene. Can be very congested on weekends.
Brownstone Brooklyn (Brooklyn Hts, Park Slope, Cobble Hill, Clinton Hill, etc.) - Very family-oriented, extremely very expensive real estate, relative low density owing to protected historic landmark status in most of these areas. Lots of restaurants, boutique shopping, etc. Brooklyn Hts has The Promenade, perhaps the best view of the Manhattan skyline you could imagine. Park Slope has Prospect Park (Central Park's bohemian, better-looking younger sister). Brownstone Bklyn is visually gorgeous with tree-lined streets, century-old Victorian townhouses, etc.
Hit up Williamsburg if you want a more youthful, energetic, stimulating experience with excellent people-watching. Do brownstone Brooklyn if you want a more relaxed stroll, peering thru brownstone bay windows to treats of period chandeliers and coffered ceilings, watching how (very well-heeled, established) NYers go about their regular lives in the loveliest urban surroundings one could imagine.
