I would highly suggest Shakespeare in the Park, i don't know which they're doing this year, but last year it was Much Ado About Nothing, and Henry V the year before that. Best of all, it's free.
As for bars and such... The Belgian Fry Bar at 123 2nd Ave (near W. 4th i think?) has the best fries you've ever had, and are connected to the Belgian Beer Bar, for some of the best beer you've ever had. Nothing like a night of beer and fries dipped in 30 different kinds of mayo, mmmm! I'm also partial to Grassroots and the Telephone bar on 2nd ave, and a place called the Sidewalk Cafe on like ave A and e. 6th. A guy named Latch shows off the Antifolk scene there, no cover, 2 drink minimum i think. I've seen some great acts there, Regina Spektor, Grey Revel, The Moldy Peaches, and Hammel on Trial. I've also seen some terrible ones, but luckily they have a pool table downstairs.
If you're looking for good food, as already mentioned Serendipity III is good, but the lines can be insanely long. There is a Dylan's Cany Bar on the corner though while you wait. I'd also recommend Mama Mexico's on 102nd and Broadway (not the midtown one, ech). Great americanized mexican food, live mariachies there too. For BBQ i'd hit Brothers BBQ, i don't remember the cross street but they're right at where Verick becomes 7th ave. If you're looking to get stone drunk on a sunday afternoon, you probably can't beat Down the Hatch at 179 w. 4th street. They at least used to run a special; $18 for three pitchers of beer and all the wings you can eat from 1-5pm.
For more cultural events, hit the museums, Natural History on 81st and CPW is dowright amazing, you can easily get lost in there. Added bonus, Bicycle Renaissance is right there too. Also make sure you go to the Bronx Zoo and MoMA.
There are literally a million more things to list, you'll never get to even a tiny portion of it in one summer. Go out, wander around, you will be amazed at what you see here, the good, the bad and the downright bizarre.