Welcome to the ‘hood!
The Petworth library is a great resource–not only for checking out materials, but they also have things like free yoga classes on weekends, book clubs, etc.
Annie’s Ace Hardware is where you’ll find many Petworth residents getting their necessities and a great starting point for getting to know folks.
Truthfully, though, I’ve met more people simply by being present in the neighborhood–walk around a lot, say hi to people, get to know the patterns of people coming and going. I like to walk around Petworth, then stop in for a treat at Flip It bakery on Georgia & Buchanan or a drink at Qualia coffee, or on a particularly nice day, trek through Rock Creek Cemetery.
I’m not a bar person, though, so can’t speak to any particular “cool places” nearby–depending on where you live in Petworth, however, you may find it easier to go to places in Columbia Heights (Meridian Pint, Wonderland) than in the immediate neighborhood.