When a waiter hands me the cork and a small taste of wine, what should I be checking for? Just that it smells like something I will like, or to check if it's gone bad, and how would I know?

I'm certainly no wine expert, but you are smelling it to see if you find it appealing. In my family people always say "if it smells good, it will taste better."

Smell, swirl the wine in the glass to check for resin/pulp, sip and roll over your pallet.
I'm not an expert on wine but I know what I like and have watched and learned. Some wines are aged in oak barrels and will absorb the flavor somewhat.