Some wonderful pics there....really reflect the vibrancy of the city. I lived there for a year, so know it quite well. Some good ideas already, but as it's your first time then definitely do the touristy main sites, they're all mostly worth it. In addition, I'd suggest taking in a game at MSQ, the Knicks aren't the best but tickets are pretty cheap.
For a flavour of NYC restaurants, I'd recommend Sparks (where Paulie Castellano was shot outside) food is pretty good but the story feels so new york, your waiter will happily tell you of mob stories of old. Also, Peasant in SoHo for Italian is a lovely place and remains popular year on year...a tough feat in the NYC restaurant business!
If you're into discount shopping then a bus ride to Woodbury common is a nice day trip out of the city....good bargains and you'll get to see some nice countryside. Bus tickets are about $20.
If the weather is kind then spend a weekend day in the park, it comes alive on the weekends.
You'll probably see a celebrity or two....my local Starbucks was near Columbus Circle and Renee Zellweger used to frequent it when she was in town (her apartment was opposite)...I had a good 20 min chat with her one day about London, horses, life and what dress to wear on Letterman...I told her the red one, which she did wear, so I like to claim that one as my call
If you have money to spend, NYC is my favourite city on the planet.