This is a pro tip, but also a summer tip, take the ferry from Lower Manhattan to Rockaway and spent the day. The beach is great and the area is impossible to gasp that it's 30 min from Manhattan (without traffic) - Eat as many tacos you can, cause it's the best in the state of New York!
Paris in Brooklyn, etc. I have to try hard not to say 'Garçon.' But either way, 'monsieur' would be more appropriate; proper French waitstaff, proper French food.
You've heard about the reading room at the New York Public Library, haven't you? Then I need not go on. Just look at it. And you're allowed to hang around in there!
Say Hi to "hip" in Greenpoint?! Yes: That´s the place for (late) summer nights and days: waterfront gastropub featuring American plates & cocktails, plus a patio with skyline views.
Weekends are crazy packed (of course).