Oh, I am so excited for you! New York is the best place in the world.
For a truly delicious meal in a casual but refined atmosphere, go to Union Square Cafe, which is on Union Square West and 16th Street. It's on the pricey side, but it would be well worth it for a special meal with your girlfriend. Union Square itself is a fun place to explore--there's the huge Barnes & Noble that Free Capitalist mentioned, a Virgin Megastore (mostly music, I can't remember if they carry books at that location), and a pretty park in the center, with lots of restaurants and other things all around. On Saturdays and Sundays the famous Green Market takes place there--tons of regional vendors with fresh, fresh produce, delicious cheeses--all the culinary delights you could ask for!
The great thing about NY (well, Manhattan, but that's pretty much the only place you'll want to go) is that is fairly small, so you can cover a lot of ground just walking.
The Statue of Liberty won't be open until August 3 (has been closed since 9/11), so I suppose that you'll just miss it.
The best thing about Times Square is that at night, it is literally as bright as daylight. You really *must* go there! The Virgin Megastore there is open 24 hours a day.
If you're interested in military history, go to the USS Intrepid, on 12th Ave. and 46th Street. You can't miss it. It's an aircraft carrier that was in operation from the 40s to the 70s, then was docked in the Hudson River and turned into a museum in the 80s. On the flight (top) deck, they have tons of military planes--fighter jets and spy planes--and some helicopters, spanning the decades. Inside there are lots of exhibits, flight simulators, you get the idea. Also on the same pier is a military submarine. When you're on the flight deck, you get an awesome view of the river and of the city. Oh, here's the website:
http://www.intrepidmuseum.org/
Oh, and while I'm thinking of it--one New York specialty that you must try is a real New York bagel! Get it with scallion cream cheese and lox (smoked salmon). H&H is a great one to try (a small chain)--there's one on 46th street, just across 12th ave from the Intrepid.
Consider taking a horse and carriage ride--you can find them all around Central Park--or a gondola ride on the lake in Central Park, or rent a boat on that lake (Loeb boathouse, on the east side between 74th and 75th; website:
http://www.centralparknyc.org/virtualpark/.../loebboathouse). Picnic in Central Park, especially on the weekend.
Especially in Greenwich Village, you'll find restaurants with small but beautiful gardens, really nice to eat in on a summer evening. For a really good guide to restaurants in NY (yes, I love food!) get the Zagat guide, a narrow, dark-red book you can find in any bookstore there. It's also online at www.zagat.com, but you need a subscription to see the reviews.
Enjoy!
EDITED to fix two minor errors.