I don't think you'll have terribly much luck getting hired, even at minimum wage, for the purposes of "learning how to do stuff." You might get a gofer job, but... eh. Those suck.
Between YouTube, internet forums, and the library, you should be able to learn the basics for a lot, and then work out the details on some small hobby size projects.
1. What skills have you found to be valuable that would be good candidates?
Automotive work is huge for money savings. You can do pretty much anything yourself with a small investment in tools (for a modern car, you need some way to read OBD-II, and I'm a fan of the Bluetooth + phone/tablet combo, since you can normally do live datalogging with those to identify issues when running under load), and for common-ish cars, there are forums filled with people who do their own work.
Not only does it save you money in terms of stuff you can do yourself, it saves you in that you know what you're talking about when you take it to a shop, and can decide for yourself what's worth it. If I take a vehicle to a shop, it's not a case of "Um... it's doing something weird, can you fix it?" I know what the issue is (or at least the symptoms and the likely fix), I probably know what part needs replacing, and I have a pretty good clue as to what the work involved is. Sometimes, I'm happy to pay someone to deal with the problem in a nice warm shop - last winter, I had a shop do the vehicle speed sensor in my truck, and weld up some new exhaust pipe while I was in there. It was that, or spend a long time cussing on my back in the snow. I was happy to cut them a check.
The best way to get started here is to buy a rolling wreck and fix it up. Your local junkyard may have a row of "Runs & drives" problem children out front, or you can look for sub-$1000 cars on Craiglist. Get one, diagnose the problems, fix it up, drive it for a while, sell it. Done properly, this can lead to a stream of free driving (I did that for a while post-college when I was flat broke, but had time on my hands).
3. Has anyone here done anything like this and have advice to offer?
On the car front, I learned to do the work because it was either that or walk, back in college. I'm often mistaken for a "car guy" - I'm really not, I just have owned a wide variety of vehicles over the years, have worked on pretty much every single one of them, often extensively, and have a good understanding of how they work and how things go wrong. I don't care to count how many nights I was up late, with the glow of shop lamps, working on a car so I had at least one vehicle running to get to work the next day (and, no, at the time, I couldn't bike, because I had to make customer site visits, often hauling computers).
I fundamentally disagree with your "I want to get a job learning XYZ" approach. I really think the right route is, "I want to learn XYZ - so I'm going to start doing XYZ." I built myself an office in a Tuff-Shed a year and a half ago, and I knew very little of the skills that went into it. Built it, learned as I went, and there are certainly some rough edges, but I learned as I went and could build a much, much nicer office out of a shed now. Just dive in, buy some tools, and go for it.