What should I check when troubleshooting a doorbell?
From the description of odd voltage readings, and the fact that you shorted the connection and still couldn't get the bell to ring, I'm guessing the transformer failed. It wouldn't be the door bell button or that wiring since you eliminated that with the shorting. And since you later tested with another power source and verified that the door bell chimes work, that's also eliminated as a possible failure. The only other thing except for the transformer it could be is the wiring from the transformer to your door bell chimes, but that can be a lot more difficult to repair, depending on how far apart the transformer and chimes are. You can double check the wiring by disconnecting it, shorting one side, and checking the ohm reading with your multi-meter on the other side.

You can pick one up relatively inexpensively from the home improvement store, so I'd start there. Make sure to turn the breaker off before touching anything that's connected to the house wiring, and always test the wires for voltage before touching them. I wouldn't bother with repairing the transformer, they're easy and cheap to replace, but hard to repair.