Do I have to wait for the reg papers to come in the mail before I go out to shoot it? That would suck. Also should I be cleaning the rifle once I get it? Just the bore with some patches or will there be stuff on the outside I should clean off too?
You do not need the registration papers to arrive, before shooting. As long as you have the printout from CFC containing the reg. number, or the docs that the gun store gave you.
On cleaning, .22 barrels/bores do not need to be cleaned after every range trip. Clean them only when accuracy deteriorates appreciably. Use only one piece rods, cloth patches and a bore guide to prevent unnecessary wear to the bore from the rod touching. Clean bolt guns from the breech, but semi autos usually need to be cleaned from the muzzle, hence the need for bore guide. I've shot more than 500 rounds between cleanings with no detrimental effect.
Cleaning too often just wears out your bore prematurely, in my opinion. As well, newly cleaned bores do not group well until some fouling has been deposited.
New barrels need cleaning just to make sure you do not have any obstructions and to clean out any excess lubricant/preservative in the bore. So go ahead and clean your new bore. Make sure it is dry when you shoot.
Wipe down the external surfaces with light solvent like kerosene, then wipe down all metal surfaces with any gun oil dampened soft non-abrasive cloth. I'd spray the bolt areas that slide within the receiver with a very light coat of gun lube as well, and let the solvent evaporate.