What type of copper pipe do I need for a typical LP gas range?
I'm not sure exactly where this pipe is, or how far back the pipe runs. Though most codes now call for a shutoff valve be installed for each appliance, in the same room and within 6' of the appliance.  I'm going to assume that the piece of pipe you're looking to replace, runs from the shutoff valve to the appliance (this might be a terrible assumption, but you didn't supply much detail about your situation).  If this is the case, you could use a piece of flexible copper pipe exactly the same size as the one you're replacing. Or you could use a corrugated stainless steel tubing (CSST). 

As you've seen, flexible copper is prone to kinking, especially if you're moving the stove often.  The CSST is more resilient to movement, and is typically easier for DIYers and homeowners to work with.  If you do switch to CSST, you might find that you need an adapter to connect it to the valve.  You should be able to find CSST fittings in various lengths, at your local hardware store.

If you choose to replace the pipe with flexible copper, you should shut off the gas and take the pipe with you to the store. With the pipe in hand, it should be no problem finding replacement pipe.  You might also find that the end(s) of your copper pipe are flared. If they are, you'll also need to flare the ends of the new pipe. Some stores can do this for you. If they can't, you'll need a special tool to do it.

Without actually seeing the installation, it's difficult to provide any more specific information.