I was in New York July 11th-July 18th but won't try to do a full trip report as others have done so well. These are just some random thoughts that might help a first time visitor.
Don't be afraid of public transportation. I bought the 7 day Metro Card for $24 and used it like crazy. I used the buses that went up and down the avenues as my own hop on and off tour. You just have to avoid rush hours and have a good guide book so you know what you are seeing and when you want to get off and see a particular site. The subways aren't as scenic but are a quick way to get around the city during the day and early evening.
Consider staying in a hotel away from the hordes of tourists in Midtown. I stayed at the Excelsior on the Upper West Side and loved the neighborhood. It's an easy subway or bus ride anywhere in the city. The streets in the area are largely residenttial and the avenues (Columbus/Amsterrdam/Broadway) are loaded with shops, larger stores, and more dining spots than you could ever try in a year. Try the diners, coffee shops, and small restaurants that line these avenues. They are relatively inexpensive and I never had a bad meal.
Use the parks as a respite from the hectic pace of the city. I loved walking through Central Park and Riverside Park people watching and dog watching and it was all free. I also got to see a free New York Philharmonic concert in Central Park with fireworks afterward. It was fun watching people arriving with elaborate picnic setups and many were very generous to offer samples to us tourists. You can walk along the Hudson River in Riverside Park during the day and watch some Shakespeare at night.
Go see a game at Yankee Stadium before they tear it down in 2009. The new stadium is under construction nearby. It was an easy subway ride to the stadium.
If you need a Southern food fix while in New York, check out the Pink Tea Cup on Grove Street (?) in the Village. For lunch I had soup/salad/pepper steak tips on rice/collard greens/green beans/cornbread/sweet tea and jello for desert all for $10! It wasn't quite like Mama used to make as the cornbread had sugar and the tea had fake sugar and lemon flavoring but it was still a great meal.
Strike up conversations with people you meet along the way. I was with a friend who was attending meetings during some days so I was on my own for part of the time and found people to be friendlier than I thought they would be. New Yorkers love their city and seemed happy to make suggestions for places to go and places to eat.