How can I learn how to fix equipment

If you're "handy" with electrical/mechanical stuff I would say you can teach yourself. By handy, I mean you generally have an aptitude for diagnosing problems with mechanical/electrical stuff, disassembling/reassembling, soldering, etc.
There are books on electrical circuits, repairing electronics, building, and testing electroni circuits that you'll need to read. I suggest your FIRST read about safety with electonic devices before you stick your hand inside one - they can get you even when unplugged. Then do as Elizabeth says - get a bunch of broken stuff and go to work.

The only thing I would say is that getting to be able to fix simple, obvious stuff is easy, being able to diagnose a problem with a complex circuit is a lot more difficult. You also need tools and test equipment, which you need to learn how to use.

Too bad that Heathkit went out of business.

There are quite a few nice audio kits around. Building one will teach you to read a schematic, to solder well, and to troubleshoot. Also you can get a chance to learn about how different components sound (e.g. different types of capacitors, wire, tubes, etc.).

I'm sure you can find one or more that will interest you. For instance, some of the upgraded dynaco clone kits are fairly easy and fun, and produce products that you may want to keep.

There are also quite a bunch of books on topics such as tube amplifier design, or troubleshooting digital circuits that you might be able to find in your local library. I agree with Elizabeth, that there's nothing like getting your hands dirty trying to fool with someone else's broken stuff, but it's important to work with equipment that you have a good chance of repairing. Most of the older (tube) equipment is pretty accessible for servicing, however, they all have very high voltages (500 volts plus) inside, so it's important to have a clue what you're doing so you don't fry your brains out.

There are also quite a few DIY sites on the net where you can browse and ask questions. Try DIYAudio for instance.

hope this helps!

You should definitely check out DIYaudio.com boards.

Also, there are some basic electronics sites easily found, such as this one:

http://www.bonavolta.ch/hobby/en/audio/audioel.htm