What are the things I need to know when operating an oscilloscope?
I've never damaged or heard of anyone damaging an oscilloscope.

If the scope has a 3-prong connector, don't connect the ground lead to something that isn't at Earth ground, or you'll blow some fuses and burn up the alligator clip.

If you're being stupid and floating the scope to measure something not at ground potential, don't leave an alligator clip hanging freely to brush against the chassis. :D

Don't connect it to anything greater than the voltage limit on the front of the scope (400 V).

Don't clip the probes to something and then let their weight + leverage bend and break the clip tip.  I'd like to know of better ways to do this.