Jump in and do it. Once you have a problem you need to fix, ask around, get a book, check online, and learn how to do that one task, and what tools you'll need.
Then do it.
Brakes are easy, though often tedious, but they're a good project for working on the car, and the manuals you can get at auto stores show you exactly how to do them, along with everything else on your car. Brakes are hard to get wrong.
For each project, whether car, home, yard, etc, there are many resources to get started. All you need to do is reconsider paying someone else to do it, and pay yourself. For your first few projects you'll be spending money on the right tools rather than a professional. Occasionally you'll spend more - especially if you make a mistake, or reach a point where you can't finish it and then have to call a professional. Over time, though, you'll learn a lot, and feel a lot more comfortable doing the work.
The big tradeoff is whether you have time or not. It'll take you many hours to finish even simple projects, especially in the beginning, but even as you get good at it there will be unexpected issues and problems to deal with. So while you may save money, you'll be spending a great deal of time. For some people the project is fun and worth it, and others do it once, then say, "Ok, I know how to do it, and I'm fine letting a pro handle it from here on out."