Nels -
I'd be willing to betcha anything that your meter is indeed internally fused. Shucks, even the Harbor Freight giveaway meters have a fuse inside. You can find it by removing the back cover (not just the battery cover). And don't sweat that you don't have a Cadillac class meter. Any multimeter that can find AC and CD voltages and resistances can be used to diagnose a motor. I do own a good Tektronics meter, but use a Harbor Freight giveaway for most of what I do. And of course, once you get started with the meter you have, you can draw your own conclusions about upgrading.
Disclaimer - I haven't looked at the videos, so I may be terribly redundant here ... but then again, we're talking about some pretty basic stuff.
First, be SURE the motor is isolated from all possible sources of power. Then use a screwdriver or something else with an insulated handle to short across any capacitors.
Switch to a resistance (ohm, Ω) scale. For simple shorts testing, the highest scale is best. For checking windings & continuity, use the lowest scale. Some meters are auto-ranging, so just set to ohms.
SHORTS
Set the meter to the highest available scale. Note what indication the meter gives when the test leads are not in contact, so you'll recognize it during testing. We'll call this "infinite resistance."
Now hold both leads against a bare metal part of the motor frame. You should see something like zero resistance. If not, you may need to scrape through paint, clear lacquer, or crud to make good contact with the metal.
Keep one lead against the metal (or even better, held in place under the head of a screw). Now touch it to each the wires and terminals you can find. You should NOT see zero or small values of resistance anywhere. If so, you have a dangerous short to ground.
Capacitors can be given a very basic "proof of life" test (AFTER THEY'VE BEEN SHORTED to be sure there's no high voltage juice left inside). Place the test leads across the terminals. A good capacitor will very briefly show a somewhat low resistance, then go up to infinite resistance. What you're seeing is the test current from the meter charging up the capacitor. As the capacitor is charged, it no longer accepts current. No current flowing in the leads = high resistance. You can repeat the test a few times by shorting out the capacitor between tests.
OPENS/CONTINUITY
Set to a low resistance scale. Hold both leads together. Note the indicated resistance. It should be a small value or zero. If not zero, this indicates the resistance of the leads and internal stuff in the meter. Note this "lead resistance" for later reference. If far from zero, something is wrong.
Now touch the leads across the ends of the various coils of the motor field. This is best done after disconnecting all wires from their terminals (you DID draw a diagram of what went where, didn't you?), so that you're looking at just one coil at a time. You should get relatively small resistance values (couple ohms at most). If necessary, you can "correct" these readings by subtracting the previously measured lead resistance. Lower quality meters will also have a random factor in the equation, but you can usually get a good approximation of what's going on. Run coils may well have different resistances than start coils, etc.
If the motor has a centrifugal switch, check the resistance across it, both with the switch closed and with it (manually) opened.
Brush-type motors can be given a few additional tests. Check the brushes themselves by placing the leads on the wire leading to the brush and on the commutator segment the brush is touching. You can check the rotor windings by connecting the test leads from one brush wire to the other and slowly rotating the rotor, so that all pairs of commutators are connected in turn. Resistance should be low, but will vary a bit when the brushes are "between" commutator segments.
Knowing how the motor is wired is, of course, very helpful (or let's call it absolutely necessary) for interpreting what you find.
I haven't even touched on the voltage scales, have I? This is probably well covered in the videos. If not, just ask for clarification.