Here are two restaurant recommendations in Columbus Circle

1. Robert (at the Museum of Art and Design, 2 Columbus Circle) Has great views of Central Park and a hip/fun decor. Menu is slightly pricey ($35 entrees) but the service was great and the food good. Make a reservation and ask for a table by the window.
2. Dizzy's Club Coca Cola in the AOL Time Warner building. Great jazz music combined with very good American food. The stage is set in front of floor to ceiling windows overlooking Columbus Circle and Central Park, and the room is intimate enough that there are no bad tables. Prices are surprisingly reasonable. You must have a reservation, usually the bands do two sets, and you must be on time.

My favorite mexican restaurant in the city is Cafe Frida (two locations on Upper West Side) One has an outdoor garden in the back that is very pleasant for summer dining. Not very touristy at all.

Also, one of the nicer spots in Central Park is the Conservatory Garden, entrance at E 104th St and 5th Ave.

Finally, the Tenement Museum (103 Orchard St, Lower East Side) is one of my favorite museums. The docents are superb, and the tours give you a real feel for what it was like to live in NYC as an immigrant in the 19th or early 20th centuries.

Enjoy your trip, there is a ton to see and do in NYC outside of Times Square and the usual tourist attractions.