To add to what Tom said, get a voltmeter (DMM). You can get a basic cheap one at Harbor Freight. The one I linked will do everything you need (and more).

It includes an ohmmeter, which measures continuity and resistance. On a low scale (20 ohms), you should be able to measure the bulb resistance. The filament in the bulb should be a low resistance (probably 5 ohms or so). If it reads open, then the bulb is burned out. This is assuming a standard incandescent light bulb. An LED bulb is a diode, which conducts in one direction only.

Using the ohmmeter, you can also check the continuity and resistance of the complete wiring system. For example, you could measure the disconnected wires, wire to wire. An open reading will indicate a break somewhere in the system. Move on from there until you find the culprit(s).