What to check for taillights not working when bulbs was replaced?

it may be a poor earth (or in the US, ground). Try to look at the wires as they go to the lights. See if you can follow them to somewhere where they are bolted or screwed into the body. It will take some effort and it would help if you had a meter which could show how far the wires were live. That both bulbs don't work suggests that the fault may be some way from the lights themselves. Corrosion is probably at the route of it. Clean as many connections as you can around the light areas.

Mike
May 2010