The first thing you need to figure out about living in New York is that your social life will revolve around the dog park. Before she found a dry cleaners, my 20-something mom located the nearest dog park in our West Village neighborhood. This is where I go to play, relax, meet friends, and get really tired so that I sleep the rest of the day.
While it's not the closest dog run to where I live, my personal favorite is Tompkins Square Park. There's a separate section just for small dogs, but I prefer to run with the big guys. When spring comes, I hope to spend most of my weekends here.