You have many options with regard to financing, but let's concentrate on the notion that you like to pay cash. I do not know how old you are and what your financial means are so I will work from an example.
Three years ago my nephew Jason, a college freshman at the time, decided he wanted to learn how to fly. He did not want to go into any debt doing it nor did he want to enter an aeronautical university with a flight program.
We put together a set of assumptions about the costs involved in flying.
1. He wanted to finance the cost of the Private Pilot Certificate himself.
2. He wanted his certificate in hand by senior year.
3. He would fly at least 2 days a week with an instructor.
4. He found an instructor who would work with him at a rate of $35 an hour.
5. He could rent an airplane with fuel (known as wet) for $65 an hour.
6. He might be able to finish in 65 hours (actually took 72).
7. He would need 50 hours of instruction excluding required solo flights and solo practice (actually took 55).
8. Books and supplies would be $200.
9. Headset would be $100.
10. Knowledge exam fee $100.
11. Designated Pilot Examiner (DPE) fee $350.
Jason decided he could work 5 days a week part time to save for his lessons. This came to about 260 days of work a year. He also realized that he would be paying tax on his earnings, this extended the work days to 325, the equivalent of 25% tax rate to net what he needed. This is the equivalent to 5 calendar quarters.
Federal minimum wage was $7.25 an hour.
He would need to work 3 hours a day to earn enough money to pay for his lessons (transportation to and from lessons, food, water not included here). Jason began saving and after 3 months he contracted to start instruction. Jason completed his rating (while taking a full class load) in just over 18 months near the end of his sophmore year.
The 3 month head start kept him on track to finance everything on his own. Originally he planned to save everything in advance and realized it was not necessary.
Keep in mind this model will change if you can earn more per hour, get different rates for instruction and airplane rental, decide to complete the Sport Pilot versus Private Pilot (take about 25% off the above Total of $7335)...
There are many instructors/flight schools that will offer the ability to buy training time in advance. I caution you against this. While they will offer discounts for advance block payments, you cannot be guaranteed the company will be around when it comes time to use the balance on your account. Pay as you go is probably the best way.
Do not throw away your investment by being inconsistent with your flying schedule. If you go twice a week for three weeks, then miss a month, you may end up having to repeat lessons because the skills you learned eroded during your time off. If you stay with your program, and have an instructor that is committed to helping you do that by being available, you will save money in the long run. Good weather where you fly will also help you get in regular flying year round. Another option is to save most of what is needed, then go to a school that helps you obtain your certificate on a fast track. These can sometimes be completed in three weeks. You are not guaranteed to be certified in that time, but you can get a great deal done over a short period.
Again, not knowing your age, credit worthiness, etc., I cannot advise you on structured finance options, but continue to ask around if a cash plan like this does not work for you.
Have fun and get out and fly,