Brooklyn Academy of Music, the Brooklyn Museum of Art, the Botanical Gardens (even this time of year!) The neighborhood of Park Slope has great shopping and dining. Prospect Park itself is bigger and has more activities than Central Park.
Williamsburg, although it’s gotten rather snobby, has some good bars and food – and expensive shopping. There are some underground, and illegal, bars along Kent Avenue. Just look for random crowds of people standing outside of a warehouse on a Saturday night.
I could go on and on – but pick up that magazine TimeOut NewYork, and a Villiage Voice (free), or an L Magazine (also free). Have fun!