In late June you might find the Central Park summer stage and events elsewhere in the other parks have already kicked off. These are usually free (or dirt cheap) concerts and cultural events that give a good feel for the city. You'll find some huge names playing, and then some local artists too. You should be able to find something on most nights. Not sure what the weather will be like in June, but July and August gets incredibly hot and sweaty and uncomfortable. Check out the city park foundation website or search Central Park summer stage for an events list, but they're great ways to spend the evening. My summer highlight.
Use the subway to travel, and walk. Depending on how long you are there for get an unlimited ride pass for 1 week.
I you're a foodie you'll be spoiled for choice. Avoid the touristy ones as you'll find cheaper better places, especially outside of Manhatten. Except for Katz's deli; yeah it's 15 bucks for one sandwich but its still the best sandwich I've had. Fette Sau is another personal fav.
You know what the touristy sights are, and most can be done on the cheap. Statten island ferry springs to mind, to see Liberty, unless you plan on visiting Ellis island. You won't be stuck for things to do, especially in the summer when there will be a lot on. But if so there are places within a days trip (Boston, DC) if you don't mind waking up early and getting back late.
Edit: I'm not a New Yorker, nor an american. I'm just fortunate to be able to spend most summers in New York.
I can't recommend any good hotels, but if you plan on doing most of the tourist stuff you're best of staying in Manhattan, just because that's where everything is. Although that's not to say there isn't a lot to see/do in the other boroughs.
Use your common sense. 5th Avenue and Times Square are hotspots for people targeting tourists. If a guy shoves a CD in your hand and starts talking "You're on MTV what's your name? blahblah" he's likely going to ask for some cash in return. Just be polite and insistent. I take cash ($), cards and a spare wallet. I haven't ever felt unsafe in New York but like any tourist centre and big city it happens. Again common sense comes in to play.
What airport are you flying to? It might be cheaper to use the sky train if you are flying to JFK to get to where you need to get to, but you might want to organise a taxi just for ease of use. Do your research to avoid getting ripped off.
And check your visa requirements. It might be you only need an ESTA which can be done online within a day, but check nonetheless. Not everyone is as privileged as us Europeans and able to travel without prior permission.