A normal prime for the D60 would be a 35mm. The 50mm would be a good portrait lens. There are plenty of options out there for either, but I'd suggest looking for one that is at least an f1.8, to give yourself some ability to work in low light, since neither of your two zooms is very fast. You also want the lens to be able to take advantage of the D60's autofocus mechanism.
You will find a good list of compatible Nikon lenses for the D60 on Ken Rockewell's excellent site. He indicates that those marked AF-S or AF-I will work. Lenses of other manufactures will also work, of course, if you check for compatibility.
If you look at reputable sources for used lenses, you can find many great values that should serve you well. But one last suggestion: good glass will improve your image quality and outlast your camera body. So think about something that you can take with you to your next camera, when you are done with the D60 and ready to try something new.