How quickly you advance depends on a lot of factors: you, the car, your instructors and club rules. Many clubs use the SLIP system to rate drivers:
https://drivingevals.com/welcome. Per this system, you should probably be 'Advanced' with at least 15 days of experience. I'd say the experience at a particular track matters as well, as this can affect your pace relative to that of your run group's, which is important.
Typically it doesn't get much cheaper without an instructor, particularly relative to the amount you're spending on maintenance and Expendables.
All that said, there are clubs that don't require instructors at all, however I'm wary of such events unless they have some pre-reqs for the drivers. Track driving can be overwhelming to new drivers, so it is beneficial not only to the individual driver, but to the group, to have an instructor in the car to assist at the beginning stages.