I just wanted to follow up on this for the benefit of any other AA newbies who might stumble across this thread.
I did make it home early, with standby on both of my flights. However, on both of them I almost didn't make it, for two different reasons, and I learned a bit in the process.
The first time, I was put on the standby list at the Admirals Club while boarding was already in progress. I ran to the gate (practically right outside the AC), and found that the agent was pulling people from the end of the standby list, with me still showing uncleared at the top. When I brought this up with him, he said he must have already called me, but I'm sure he didn't because I would have been able to hear him on my way over there. I think he just didn't look back up to the top of the list after he'd passed it. In any case, only by getting his attention was I able to snag the very last standby seat on the flight.
On the second flight, there were several people ahead of me on the list, and they called them out the normal way over the PA. Then there was a long delay, and I figured they had finished calling people (strangely the list wasn't showing at all on the screen. I only knew people were ahead of me because the AC agent told me). Another passenger who had been waiting went up to the desk and talked to the agent, from what I could hear, trying to find out if there was any way he could get on the flight. Shortly thereafter, the agent said my name, almost casually, without the PA. Only because I was standing less than 5 feet away was I able to hear him, and go up to get my boarding pass. I try not to be too cynical, but it kind of seemed like he was trying to get the guy in front of him a seat by having me not hear my name and then passing me over...
In any case, I learned: always stay near enough to the desk to hear what's going on, and speak up immediately if it looks like they've passed you over on the list.