The most important debugging tool in any E.E.'s toolbox is a trusty multimeter. A multimeter can measure continuity, resistance, voltage and sometimes even current, capacitance, temperature, etc. It's a swiss army knife for geeks!
You should go through all of these sections in order, as they build on each other.
Everyone always asks What multimeter should I get?
Well, since they're rather commodified (there are dozens of manufacturers) it is hard to make everyone use the same model, even though it would make things easier.
These are the necessities:
Here are nice things to have in your meter
These are things that I rarely (if ever) use, in decending order
I have a selection of suggested meters on this page, I strongly suggest the "better" (At $15 its a great value) or "best" ($50, autoranging, has a temp probe, all sorts of other junk) ones as the "good" one is really kinda bad. If you already have a multimeter then just use that one. I'll only be covering digital multimeters here so if you have an analog one, it may be a little different and you'll have to experiment to figure out what's up
A good multimeter will cost about $25. I suggest getting a ranging one , with current testing, and a wide range as well as easy-to-replace battery. You don't need to spend $200 on a Fluke 73! A $50 meter will be excellent. Auto-ranging meters may keep you from tripping up on ranges at first, but they're slower and often flakier.
I have had a Wavetek Meterman 27XT for half a dozen years and its been really good to me, but it was also about $80.
Here are many other links for learning how to use a multimeter