What's your budget?
That'll probably be the decided fairly quickly.
I've got a D80 & my mate has got a D60. I think he got his with 18-55 kit lens for £350, which is seriously good value.
Whenever I have used his camera, the only things that I prefer about my D80 is that is is a bit chunkier & easier to hold – it feels better balanced with a decent telephoto on & you can change things easier on the D80, as most things are available through a button push and twiddle of a jog dial. On the D60, I think you have to delve into the menu a bit more.
I'm sure that once you are used to it though, it is not that much slower.
Also the D80 has and LCD on the top so you can use that instead of having to fire up the main display. Not sure if that's better or not, just different.
Oh, on the D80 you can choose to record RAW + jpeg at the same time at all levels of compression. With the D60 it's Raw, plus Basic jpeg only.
And the D60 won't work as a master controller for remote flashes. So, if you see yourself getting into flash photgrapy more (using remote flash units) then if you get the D60 you'll have to go for the top of the range Nikon flash or get a separate commander unit.
I'd say if you aren't gonna get crazy serious with the whole photography thing, then the D60 will be great. If you intend to get a bit more creative in the future, then the D90 might be a better bet. (Not so sure about the new D5000).