Will my lenses for my nikon d60 fit on a Nikon d5000?

I'm looking somewhat upgrade my camera body, i have a nikon d60 and 3 lenses that fit it, will they fit a Nikon d5000?

Of course. Nikons can use any lens made since 1959 (some may not be able to use certain features, though, (like the D40 won't meter with older lenses).

In short, yes.

the Nikon D5000 takes pretty much any F-bayonet mount lens, but can only autofocus with AF-S or AF-I lenses

Yes. Yes. Yes. (for all 3 lenses)

The D5000, like the D40, D40x, D60, D3000, new D3100 and the newer D5100, have exactly the same lens mount and autofocus capabilities of your D60. Any lense that works on any of those cameras will work on any of the others.

Here's a list of fully compatible lenses for future reference.
http://en.wikipedia.org/...ocus-motor

Yes. In fact, a Nikon lens made in 1960 can be used on the D5000 if you don't mind manual focus.