There are literally hundreds of different airplane types you can train in, although Cessna and Piper do seem to have a big market share. Getting a PPL will give you an SEP rating, which means you can pretty much fly any other similar single engine piston airplane (with a few exceptions requiring extra training). A bit like taking your test in a Ford then buying a Vauxhall. It's always a good idea to get 'checked out' on a new airplane by an instructor, and if you're renting they'll insist anyway.