I have the D60 with a VR lens. This combination does pretty well in low light on things that don't move too much:

1. The VR on the lens allows you to use slow shutter speeds without getting blurring due to camera shake. My most extreme example so far is 1/15s at 85mm (normally I would not go below 1/125s).
Note that this shutter speed is too slow to stop a moving subject. If you have to shoot a moving subject then as Frog says a real low-light lens like the 50mm f1.8 is a necessity (but it won't autofocus with the D60 - look rather at the D80 or D90)
There is now a VR version of the basic 18-55 lens, which is probably enough for your needs. Check if you can find a D60 in a kit with this lens.

2. The D60 has the possibility to use Active DLighting in contrasty situations. This reduces the exposure to avoid losing the highlights and brightens the mid-tones to compensate

The D60 and the D80 have the same sensor but different image processor. I find the D60 is very good at 1600ISO. It looks like 400ISO film..