$300-400 for a good condition D60 is about the top reasonable price. The quality used vendors (i.e. KEH) are selling the bodies in this price range. You didn't mention what lens is being included(?) and that can make a huge difference in cost with this level of DSLR as the lens may multiply the cost 2 times or more.

Financially you are most likely not going 'up' unless it is really a bargain and the D60 and lens are in great shape.

As already mentioned a DSLR will not give you better photographs all by itself. The pictures may be marginally better initially technically. Less noise, better overall exposure, etc. Your composition and selection skills will still be the same.

Keep in mind that a new Nikon D3000 with a 18-55mm lens is currently selling for $500. A new D5000, which includes 720p video capability, is selling for around $700 with a VR (Vibration Reduction) lens.

There is Photo John's Best Entry-Level Digital SLR Guide to use as a resource. Several suggestions in this article about options and consideration in buying a DSLR camera.

If the photo bug has really caught you, or you take a lot of family photographs or are going on a really special vacation trip/tour, consider investing in a new DSLR. A current entry level DSLR is capable of better quality images than all but a handful of what are considered Point and Shoot cameras.

Do keep in mind that the DSLR is rarely as convenient to carry/use for most people. They are bigger and bulkier and can as easily screw up a photo as not when used carelessly. After all, its the camera that you want to use that will ultimately take the most photos you will keep.