There are two kinds of training.
The first and most important is "Crew Grade" training. This affects the crew grade you see in the fleet summary window. It's a percentage bonus to a whole bunch of things. Off the top of my head, it increases to hit chance for weapons, improves maintenance (?), improves damage control, and reduces jump shock duration.
The crew training level of your academy affects the starting value for crew grade (at training level 5, you start at 12%). The max is 22%. It increases over time based on the full bonus of your executive officer (assigned to an auxiliary control) and half your captain, plus a bit of the bonus of any admin commanders if you are in a naval or training command.
The second kind of training is fleet training, which starts at 0% and slowly increases to 100%. This is what affects order delays and stuff. It only applies if you have inexperienced fleet penalties turned on, and is also mitigated by the reaction bonus of the ship CO, the officer assigned to the fleet flag bridge (if any), and a bit of the reaction bonus of admin commands (I think naval and patrol).
Fleet training naturally accumulates according to the crew grade. It is ALSO increased by the crew training bonus of officers in a training command hierarchy, though ships in a training command burn fuel, deployment time, and maintenance life at an increased rate. It is NOT directly affected by the training bonus of the actual ship officers, though having high training bonus officers means you get to 22% grade bonus sooner.
Mechanics questions are best answered by reviewing the C# changes thread, which has the actual mechanics written out. There is a lovely table of contents here:
http://aurora2.pentarch.org/index.php?topic=10666.0I have it bookmarked.
The specific post about crew grade is here:
http://aurora2.pentarch.org/index.php?topic=8495.msg103818#msg103818Fleet training is here:
http://aurora2.pentarch.org/index.php?topic=8495.msg109158#msg109158EDIT: I never actually talked about the Aux, my bad. I haven't done the math, but I've noticed in my current conventional start campaign that my ships reach max crew grade well before finishing fleet training. This is with 4 stacked training commands with officers with 225+ crew training bonus. I haven't researched Aux yet. Since I'm not gonna deploy the ships without max fleet training anyways, I'm not sure the Aux is worth it. Especially since, once the crew is fully trained, the Aux and executive officer are dead weight.
Obviously adding the aux would make the ships "work up" a little faster, which could be useful in a pinch. But it's a fairly large investment for what amounts to getting a 7% bonus (8%/112% starting value) a little earlier. Especially with how hard it is to have enough officers for all your ships.