What do you need to do to accompany a minor (16) year old to the gate and pick them up from their return flight at the gate?

Check at the ticket counter

Check at the airline ticket counter. They should be able to provide you with a gate pass.

Matthew Fultz has some good advice. Your best bet is to call ahead to the airline. Typically, only ticketed passengers can pass through security and minors are escorted by airline personnel but policies differ from airport to airport and TSA's rules can change as well. It's best to call ahead.

I would definitely call the airline they are flying on. They would be able to give you the best information for that. Most of the time, an employee of the airline/TSA will walk the minor through security and will walk the minor to you at the none TSA secure portion of the airport (usually the same place everyone goes after flight arrival, like a pickup location). You normally cannot get past security checkpoints if you are not a ticketed traveler. You fill out a form listing who the minor is to be released to and that person is required to have a photo ID in order to "take possession" of the minor from airline staff.