I'm a native Lawn Guylinder (Long Islander), worked in the city when it wasn't great, and visit quite frequently nowadays. One of our favorite restaurants in Manhattan is right across from Lincoln Center. It's upscale and called Rosa Mexicana. I highly recommend it. Gil could give you more choices as he used to live around that area. Others we love are Vong, Sushi Samba, and Dylan Prime.
One comment? Anne suggested it's only safe south of Central Park south. I would like to add if you're on the upper west side up to 72nd street or on the upper east side up to 90th street you'll be fine also. Just stay out of the park after dark like suggested (but be sure to go during the day!) Also the whole east side to the river is safe. Even Times square is safe at night if you stay aware of your surroundings, but I'd give that advice anywhere in the city.
If you cab it or bus it downtown, the village is alot of fun at night and safe. Also try chinatown and/or little italy.
You'll be fine wearing nice slacks and a sports shirt for any of the better restaurants, (there are a few exceptions where you might need a jacket). Definitely do yourself a favor and get over to see the Chagall windows in Lincoln Center and maybe catch a concert or a ballet, depending on what's playing then. Also, one night you should consider getting show tickets.