I would not opt for the P22, it has a very goofy manual of arms (manner in which the gun is operated by the user), the Browning .22 would also not be a very good choice in my opinion as it lacks a lot of good traits found on a target gun. The P22 actually requires the trigger to be pulled to de-cock the hammer after engaging the safety....if you forget that part, you may end up with a book instead of a click. The Browning 22 1911 is a reduced size copy of a 1911 GI model and has pretty stubby sights and for a good target gun you'll want better.
The Browning Buckmark , Ruger 22/45 and MkIII pistols are probably the three most popular lines of target and plinking pistols and I would recommend on of those three. Possibly the new Ruger SR22 as well.
I think starting off with the .22 is a good idea. You can get a lot of the fundamentals down with that and think about what you want in a center-fire option later. If you are wanting to get into USPSA etc. Once you get some more trigger time start hanging out at matches and such and see what they guys are shooting. Contrary to popular belief, you don't need a tricked out race gun to compete in USPSA as there is a "Production" division that allows normal guns to compete against normal guns.
Also look into Steel Challenge matchs and Ruger Rim-Fire matches around your area.