The Museum of the American Indian is always free (it’s a Smithsonian institution). This is large, and could take a full morning or afternoon.
The Forbes Gallery on 5th Ave. and 12th St. is one of those small, quirky collections of neat stuff (antique toys and maps, that sort of thing) and it’s also free. This is more of a drop-by.
The main New York Public Library (with the lions) in mid-town is also free, and neat to walk around inside. They have gallery space on the upper floors, with rotating exhibits.
The walk up 5th Avenue is that classic New York shopping corridor – I usually advise people to start at 50th, at St. Patrick’s Cathedral and go up to 59th, which is the park. This walk gives you all the big name stores, and works (IMHO) equally well whether you are a person who likes window shopping, or if you are a person who likes to be appalled by gross commercialism (in that “it’s fun to be scandalized” sense). If you are a more sturdy walker, you could start at the library on 42nd, and go north from there.
Central Park, of course, is free to wander around in, and there are a lot of things to see.
Rockefeller Center has these neat self-guided map/tour booklets, you can pick one up at the info center in the lobby and they’re really quite informative.
The Union Square Greenmarket is probably my favorite market in the city, obviously the things at the booths cost money, but a cider doughnut is probably worth it. Most of the vendors are fairly local and it’s an interesting look at what’s going on with organic/eco-friendly food in an urban setting. The Union Square Partnership sponsors a free walking tour of the park on Saturday afternoons.
The Museum of Modern Art (MOMA) is free on Friday nights. It does max out and they stop admitting when they hit capacity, so early might be better. The American Folk Art Museum is right in that same neighborhood (midtown) and also free on Friday evenings.
You can take the Roosevelt Island tram for $2, and is a neat view, unless you don’t like heights and/or going over water (my mother, when this was proposed to her, said she would pay $2 NOT to take the tram). Once you get to Roosevelt Island, there is not very much to do/see so the point is to turn around and go back. If you do this, do it in the daytime, the area gets a little desolate at night. In other public transportation news, the Staten Island Ferry is free, so one can ride it to Staten Island and back (last I did this, you had to disembark and get back on). This time of year, I would say this is a nice thing to do if you have a great clear weather day, but not if the weather is overcast or raining or insanely cold.