Yes, you will save money training in your own plane, IF, and only IF, you plan everything well and luck is in your favor.
Seek something that has a proven track record like a Cessna 150/152 or a cherokee 140. You will be able to operate these birds for around $65 an hour. You can actually pick up a Cessna 150 for around $15,000. Because these planes hold their value well and are always in demand, once you have all your ratings you can sell and upgrade to something larger like a 182 or Arrow. It's a great option, but requires careful planning.