Looking back, I wish I had purchased my own aircraft to do most of my flight training in.
Assuming you are getting your Private Pilot Certificate, Instrument Rating and Commercial Certificates you could very well save money by purchasing your own trainer.

An economical trainer will burn around 8 gal/hr. In my area we pay approximately $5/gal of Avgas. This comes to an hourly operating cost of about $40/hr. An average trainer in my area costs about $125/hr to rent. Based over 250 hours of flight you stand to save upwards of $20,000 by owning your own aircraft.

The above comparison is a very crude comparison, you still have to factor in insurance, maintenance costs, the airplanes annual cost and so on. Also, you won't be able to conduct all of your training in your aircraft as you need a minimum amount of hours in a complex aircraft when going for your Commercial Certificate, or your Multi-Engine Rating. These figures do not include the actual cost of the flight training it's self.

Once you are finished with your flight training, you can sell the aircraft to free up the money you invested into purchasing it.