It's the level of tech that I grew up with so I'm not sure where you'd learn it other than just to get your hands on one and to start digging in. Very few tools required if you stick to the test-and-replace-subsystems level, a screwdriver, a good DVM, a soldering iron (a good one, preferably one that is isolated from mains), a lot of spare time and a notebook so that you can keep track of what you find and build up a store of knowledge over time to help diagnose problems faster.

About 200K DX-7's were made, which is enough that you can find them and not so many that there is no value in repairing them, on top of that the degree to which people are attached to their synths is something that still amazes me. The level of emotion when a dead synth comes back to life with the owners is always quite gratifying, it's some of the best time spent for me.