It is important to note that your old Nikon F60 is not a "manual" camera, in fact it is an Autofocus capable unit, which does have manual controls. The F60 used the standard F mount lenses only, it was not compatible with DX lenses or the like. What that means for you, is that your F-mount lenses are all compatible with all recently made DSLR's that Nikon produces.
What you really might have to be concerned with is if you get autofocus or not. Some entry-level cameras will not have the autofocus motor built into the body, and as such you won't be able to autofocus with older lenses that do not have the motor built in either. To sum up this situation, yes the lens will mount and function, but may not autofocus with the new DSLR body depending on which one you get.
To determine if your lens/body will autofocus or not, this chart may help out:
Yes you can buy just the body and not lenses, but many of the kit lenses that are available today with DSLR cameras are likely to be of higher quality then the kit lens you received with your F60, and they are offered at a very reasonable "kit price".