I would definitely recommend renting a car.
I've never been to any other grand slam, but the US open seems to be okay. There's parking right across the street at the Mets stadium if you are going to be getting a car. Also, there's a train that'll take you literally right to the US Open from Penn Station. I personally would recommend staying in New Jersey and go to the city by train or ferry and then you can take the subways/taxis around. Even if you get a car it's easier this way because it is an absolute nightmare for traffic. If you pick the wrong time of day you can be stuck going across the bridge/through Lincoln Tunnel for hours.
New York City's pretty awesome, but if you want to get away from the city it's also beautiful. I personally love Pennsylvania. You could go to Philly, you could go to Lancaster and see the Amish, you could go to Hershey Park if you love chocolate and it's also a decent amusement park, and Knoebels which is an amazing amusement park in Elysburg!!.
Anyway, you could drive to Baltimore if you like zoos or aquariums. Washington, Dc would definitely be within a driving distance if you really wanted to go there.
It'll still be nice enough during that time if you want to go to some beaches, I wouldn't go to any beaches in New Jersey, but Maryland (especially Ocean City), Rhode Island, New York, Massachusetts (especially Cape Cod/Martha's Vineyard) will have some nice beaches. Further up north will be really good if you want to go whale watching or anything. I've been up to Cape Cod a couple of times to go whale watching and it was an amazing experience.
Theres a lot you can do in the North East, it just depends on what you want to do.
I cannot believe you mentioned Knoebel's
I used to go camping there every summer
I think it's a slight exaggeration to call it an amazing amusement park
But it is certainly amazing for the price! And the funnel cake is
and I love that ice cream at the place that has that water wheel thing.
But it is quite out of the way and not all that exciting for an overseas visitor who only has a week here
I will think about my own personal advice and type that up tomorrow.
That said, USO tickets are very easy to get
I would only worry if you plan on attending the USO over Labor Day weekend. Otherwise you'll be fine