REI mechanic here. The basic bike maintenance classes are essentially there to show you how to keep your bike in check in between tuneups. So we'll cover topics like:

basic cleaning
lubrication
tire/tube change (and brake disengagement [for the V brakes])
proper torquing
chain checking

We'll also give pointers on things we look out for during inspection before a bike is checked in for a tuneup.

I'm not sure exactly what others go over, but we all for the most part have a general guideline on what needs to be covered. It definitely wouldn't hurt to attend a class, especially if it's free. The mechanics will usually give little insider tips and tricks on making your life a lot easier as well.