I don't think there is much you could damage on a car with a meter. It is the meter that is more likely to get damaged by measuring too high a voltage, such as the ignition voltage, maybe 30 kVolts or too high a current, such as the starter current, around 100 amps. Always make sure the meter is set to a much higher range than you expect to measure. Make sure you understand how to use the meter and then practice on a few easy measurements. Try the battery and reverse the leads to see how the meter indicates negative voltages. A good place to measure a current is on the fuse panel. Remove one of the fuses and connect the meter across the fuse holder so the current will have to flow through the meter just like it did the fuse.

It's hard to do diagnostics if you don't know how things work. Get a basic book on low voltage DC theory. Learn Ohm's Law, current flow in series and parallel circuits, voltage drops, etc. Then learn how the various parts in the car, battery, motors, solenoids, relays, diodes, etc are supposed to work. Good hunting!