Very good information here from a couple of our best contributors, so I'll toss in my $0.02 worth.
You'll want to understand the basics of electric circuits, so taking a beginning basic electric theory course would be useful. You'll learn about resistors, capacitors, Ohm's Law, current, power, resistance, inductance and all sorts of mean, nasty, ugly things. No, actually, that's just a quote from 'Alice's Restaurant'; that stuff is all pretty amazing in how it works and all fits together, but you'll definitely want to learn it in order to know what's going on with the devices you're working on; as @oldturkey03 stated,
[quote]This is not about being able to microsolder a 0201 capacitor, it's about why it is on there and what it does.[/quote]
For instance, you've got a phone that's not charging. With a basic understanding of electricity and circuits, you can use a voltmeter to check to see if you're getting power to the charging port, then follow the circuit to the motherboard and verify it has power and figure out exactly where the fault is rather than just taking a shotgun approach to randomly replacing related parts like the lightning port and hoping that's the problem.
A local community college would be an excellent place to start, but there are a huge number of options open to you for learning. Online academies, self-taught tutorials, books and videos can all help your understanding.
With that knowledge, you'll have a much better grasp not only on what to fix, but also why that fix works. All of the parts are interrelated, so once you get to know one part it helps fill in the overall picture.
Anyway, I'd like to say welcome aboard! One thing about being able to fix things - you'll never run out of things to do! And if you enjoy it and can earn a living at it; well, that's my definition of success, right there!