As noted by OffGridCamper, I too grew up with it; my father taught me most of the basics when I was a kid and I learned much of it just watching and listening when he was doing something to the family car. I started by wrenching on my bicycle, then when I turned 16 I started wrenching on my car because as a family we simply didn't take things to a mechanic. I do recall a few times when Dad had one of his buddies over to help with a job, but like most men in the 70s (at least all the men I knew back then in rural Indiana) he did all his own work.
Hanyes and Chilton manuals are a great way to start. Even better is to go help a buddy or neighbor when they're doing something on one of their vehicles. You don't have to be able to much more than hand tools to the guy when you start, even that will help get you familiar with what's going on under the hood and de-mystify some of the systems. Of course today you have YouTube and forums for most models where you can discuss specific questions. Always be a little wary about the forums - you can usually spot the guys who actually know what they are saying vs the ones who are just blathering on but when your new it might be harder.
Then just get out there and start. You don't need $1000's of dollars in tools, but do buy a few decent quality things to begin with. Craftsman isn't anything like what it was 20 years ago but it's still a decent home-owner brand and generally a safe bet. The cheapest way to get started will be to pick up one of their 'xxx piece mechanic's tool sets' which will have all the common 1/4, 3/8, and 1/2 sockets plus wrenches, allen wrenches, screwdrivers, etc in both metric and standard. With Father's Day approaching they usually have some decent sales going so it's a good time to look. Used is fine as well but harder to pick out a full set and sometimes it seems CraigsList sellers want nearly new prices anyway. I do tend to stay away from the hardware store stuff as like Lowes or HD as they don't really serve the mechanic very well and their stuff usually seems poorly designed to me. Beyond that you'll want to add a good floor jack and a pair of jack stands and also a basic DVM with Volts, Ohms, and continuity check. You don't need super precision for basic stuff; even the freebies they give away at Harbor Freight will do to start.
Speaking of Harbor Freight, some of their stuff is decent and some is absolute crap. They get better every year but you still have to go in knowing what you're looking at. Once upon a time everything they had was crap but they've been steadily pushing up-market and these days I'm willing to buy some things from them like jacks, stands, a shop press, and probably their single best item, the 44" cabinet. But I don't generally buy any actual hand tools from them (again, getting better every year so maybe one day) and I never buy any sort of power tools from them. With experience you'll learn where they're strong and where they're to be avoided.
With any modern car there is one more tool you'll need that we didn't have when i was growing up - a good OBDII scanner. I use a model specific one for my Land Rover that's pretty expensive as it can do more than just read and clear codes, but the generic ones can be had fairly inexpensively.
Like any skill, it's practice, practice, practice. The first time you do something it will take 3x longer than it should. The next time 2x. Before long you'll get to know the in's and out's of your vehicle and be able to handle most of the routine maintenance. Then you'll start installing your own accessories. Then you'll want more tools. Then you'll start doing major jobs. Then you'll want more tools. Then you'll be ready to rebuild your first engine. Then you'll want more tools....
Hey, at least tool addiction isn't the worst thing in the world.
