How to learn to repair consoles?

@hhrl , it really does not matter what you want to repair. What matters is that you learn the basics. First learn to walk, then start running. One of the first things to get into would be to read schematics and how to approach proper troubleshooting. You need to learn what the components do, why they are there etc. Collect as much information as you can about the devices you are interested in. get as many guides and service manuals as possible, you'll be surprised how handy they will become. Try to find most common errors and their fixes. Seriously, that is have the battle. Videos are okay as well but remember that you learn from doing it and not just watching it. Just check on here (ifixit) the amount of gray screen issues, overheating issues of Xbox's and PS3's. That way you will not just have to go and keep replacing parts. Of course, a knowledge in basic electronics will help as well, it's nice to know the difference between a capacitor and resistor. Another really important thing, in my opinion, is to learn how to properly use a multimeter. Soldering will become another issue. Most devices are easier fixed with a hot air station due to the fact that they are using a lot of SMD components. You may also consider investing in a small reflow, or rework, station. Not very expensive but a great tool to have. Get a bunch of broken things and practice. Keep notes, pictures etc. to refer back to. Keep a journal of common errors, fixes etc. All that stuff will come in handy. There are many, many more things, but I do hope that this is a start.

I am certain that the game console guru @dushku can add way more specifics for you