Well as a former airplane owner I can say for sure that you will NOT save money by owning your own airplane during training and maybe not even after training unless you fly 200 hours per year or more.

However you might lease your airplane back to a flying club or flight school to help pay for the airplane.  Most people own airplanes so they have one available when they want not to really save money.

If I was to purchase an airplane again it would either be a Light Sport Aircraft (LSA) or a Cessna 172.  4 passenger airplanes will get more use, the C152 is only good for two lightweight people.

I had the same thought -- in fact, I had met someone in my travels who started out really concerned that the aircraft he was training in were going to get sold out from under him. This encouraged him to buy his own aircraft, and eventually he got into a business leasing them.
But time out. One quickly realizes that it just isn't so. Apparently, no one is in a rush to buy a flight school plane. They are not going anywhere anytime soon. Often you will see an ad posted for a 172 for example, and somewhere within the ad the seller will state "NEVER WAS A TRAINER" or some other hint (like posing the entire family in front of it, or stating "one owner, privately used" or something similar).
I can personally say that I've been training for several hours in the same 172 and I never feel a lack of confidence in the airworthiness of the equipment. In fact, this particular plane had suffered some serious damage in an accident 5--7 years ago. It would probably cost about $50,000 to fix it had it happened today.

This brings up another point - damage history. As one person bluntly put it, "If someone tells you that a 30-year old plane has no damage history, they are LYING." Since you are NEW to the world of flying, how do you *really* know what you're getting into?
What is the likelihood that you buy a used aircraft, then find out at the next annual or major overhaul that it needs MAJOR work? What if that "$25K plane" turns out to cost twice that much in mechanical work (hence the reason for the quick sale...)

The only thing I can determine at this point is that you would pay for a thorough mechanical inspection prior to sale. (I wouldn't use a mechanic recommended by the seller)
Not only might this protect the buyer, but I think it's good for the seller as well - he/she doesn't waste time with people who aren't serious. I think that a willingness to pay for an inspection is a clear demonstration that you are serious about making an offer.

As for ownership of a trainer -- one risk is YOU and your needs. Right now you're training and learning to fly. Pretty much if it's cheap and gets off the ground (and an instructor is willing to get into it) that would meet your needs. But what about after training?
Will it have the utility you need?
If not, what will you do - sell it?
How much of a loss will you take?
There's something to be said for renting a "wet" plane by the hour and then walking away from it after the session is over. Repairs and maintenance are not your problem.

As far as a leaseback is concerned, do you really want other students flying your plane?
What if it's damaged while out with someone else? Sure, insurance can take care of that. But while it's in the hangar, it's not available for use. So either you're not flying or you're renting something else. Doesn't this defeat the whole purpose of owning your own aircraft?