Truly you can find a great place on every corner however here are some to note. Fantastic historic steak house Peter Lugers, Fette Sau best barbeque in the city, Meatball Shop for good old comfort food, Robertas has good pizza and farm to table locally sourced ingredients, at Dumont the burger is big, juicy, and delicous, mac & cheese is made w/ bacon (enough said)!
Start your night at Hotel Delmano for gastronomic cocktails & charcuterie , Hit the Knitting Factory or Brooklyn Bowl later for live music, to keep it going & if you've got the urge to dance, head over to Union Pool for more live bands and a social scene, wrap up your night at the Commodore, a local bar with a inexpensive drinks and killer fried chicken
For a wonderful little brunch spot try Egg, for a croissant and espresso head over to Margo, a little patisserie tucked away on a quiet street. Otherwise head over to Blue Bottle for the best coffee in New York City. Other nice little cafes with free wifi and boulangeries; Bakeri, El Beit, Atlas, & Ella
If you are looking for vintage clothing or lesser known or european designers and boutique shopping you have hit the jackpot. Otherwise my place is an 8 minute train ride into Manhattan where you can visit all the usual spots, from Soho to 5th Ave, it's all here.
Bit of culture
There are galleries and live music all throughout the neighborhood. I'll supply a complete list of places to visit while here.
Walks in the park
The obvious is Central Park, however a few lesser known and equally beautiful are Prospect Park, Ft. Greene, Riverside, and Ft. Tryon
Spot of exercise
There are many yoga studios in the area. As well as a track at McCaren Park.
Catskills for silence, culture and nature, Montauk for a good beach scene and surfing, Hamptons for the summer parties and beaches, otherwise same goes for Fire Island.