If you have the money, I would recommend that you buy your own plane, but only after you solo. By then you will know for sure that you really want to pursue aviation, and that you don't have airsickness issues...(My dad, wanted so badly to be a pilot, but couldn't get past the airsickness. He flew RC planes, and got me addicted to flying at a very early age. Fortunately, I didn't inherit the motion sickness issues. )
You can learn how to fly in any plane. The US Navy's primary trainer (back in my day) was the T34C turboprop, with retracts, constant speed prop, and 210 knot cruise...I chose it because it had AC...some of my buddies flew for the first time in the T-28.
However, I don't recommend that for civilians that have a budget and aren't full time flight students...
I recommend you look at the Grumman Cheetah or if you have the budget, a Tiger. They are both FUN planes to fly, simple and very affordable. They have fixed gear and fixed pitch props, nonetheless, their performance ain't shabby. A well rigged Tiger will run with or outrun a 200 hp Piper Arrow. They aren't as "hot" in the pattern as a Mooney, but, they are more 'similar' than any other trainer that I can think of.
Get a hundred or two hours under your belt, then look at moving up to something more complex...