Using a VOM is pretty straightforward. There are three types of measurements a VOM can make (some can only make two) 1.Volts, 2.Ohms (resistance) and 3. Amps (current).
For volts, you generally select the type of volts (AC or DC) and a voltage range, on a knob or rotary switch selector. For AC volts, the selection will generally show a ~ symbol. The other volts ranges are usually DC.
For resistance (Ohms) often you select an expected resistance range. When doing continuity checks almost any range will do and you're looking for the lowest reading. The lower the better here. WARNING: Don't try to read resistance on any powered circuit. To do so will damage the VOM. Always remove the power from anything you want to test resistance or continuity on. Best is to have whatever you're testing completely disconnected, both ends if possible (but one end will do).
For current, the meter has to be in the circuit. That is you have to make a deliberate break in the circuit and use the meter to bridge that break.
The key thing with VOMs is only use the selected range for the type of measurement you're doing. To do otherwise risks damage to the meter and possibly to yourself and possibly the item being tested. Be particularly careful around AC voltages.
To test a fuse, it's easiest to remove the fuse from its holder and use the Ohms range to read across the fuse (not the holder). Again here, the lower reading is better although a fuse will likely only give you some resistance reading (good) or something like "0L" which is a blown fuse.
Electrical circuits are simple at their heart. The electricity starts at a source (AC active or battery +), then goes through some type of switch then through some kind of device, then returns to the source (AC Neutral or Battery - ).
The types of things that go wrong are:
Open circuits: A break in a wire somewhere along the path. (A faulty switch will do this too).
Short circuits: The power from the source goes directly back to the source without going through the switch and device first. This is when a bare power wire touches a bare return wire, or the frame which is connected to the return to the source (ground).
Device failure: The thing that's being powered needs to be replaced or repaired (if you can).
A switch is just a device for creating a controllable open circuit.
That's pretty much it. From that you can diagnose just about anything with a bit of logical thinking and a meter and a bit of snooping around the wiring.
Here's a tip: When you're snooping around the wiring of something, I find it helps to draw the circuit on a piece of paper for working out how it works and for later reference.