Is there a proper way to test a tube tester? I aquired a Mercury 1101c tube tester a few years ago and have never plugged it in. Are these like other things where I should warm them up somehow or replace caps before I apply power?

IIRC that is a simple emission tester with no active components inside to need warmup. Replacing any paper or electrolytic caps is a good idea, however.

As far as testing, check a selection of known good and known bad tubes and see what results you get. Emission testers aren't the best way to test a tube, but they are a bit better than nothing. If there are internal adjustments, you will need the manual to figure out how to adjust/calibrate the device.