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!
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:
Continuity testing with piezo buzzer
Resistance test down to 10 ohm (or lower) and up to 1 Megaohm (or higher)
DC voltage test down to 100mV (or lower) and up to 50V
AC voltage test down to 1V and up to 400V (or 200V in the US/Canada/Japan)
Diode testing
Here are nice things to have in your meter
Auto-off - to keep from draining the batteries
AC and DC current test, from 10mA to maybe 200mA and then also a 10A one as well
Stand - a thing that flips out and keeps it upright on your table
Auto-ranging - note: some people don't like auto-ranging because its slower and not as precise
Hold - keep the maximum value on the screen so you can probe without looking at the meter.
Common battery - such as 9V or AA's, pocket meters use hard-to-replace coin cells
These are things that I rarely (if ever) use, in decending order
Frequency counter - before I had a scope this was surprisingly useful!
Capacitance testing - usually to check random SMT caps
Inductance testing - how often do you really use an inductor?
Duty cycle - never used this
Transistor beta meter - people don't really work with transistors anymore
Temperature probe - I use the "Pease temperature test": a finger