Question: It has become clear that I need to practice my transfer techniques; I just completed a primary-to-secondary transfer which involved what I'm guessing is a fairly significant amount of aeration. During my transfer (1-gallon basic brown beer, direct from carboy to carboy; no intermediate pot), I used an auto-siphon, and ended up pumping oxygen in. In addition, there was air in the line even while the beer was flowing (which I had to restart several times).

Ultimately, if a beer gets aerated, is it a solid bet that the beer will be oxidized and pick up that "musty, cardboard, stale odor and flavor"? Should I not get my hopes up, or might things still work out?

**Update:**

Last night, I opened one of the bottles. Much to my surprise, it was *not* awful! The carbonation was a little low, it could have had more flavor, and it was a *little* stale, but it was not the disaster that I was expecting. Hooray!

Answer: 
The bubbles are large so although some oxygen will have been absorbed it's not as much as say sloshing the beer around in the carboy for 60 seconds. 

The beer will probably oxidize in time, say 2-4 weeks, but you can do two things to mitigate this:

1. drink the beer within a couple of weeks
2. store the beer at fridge temp. This will reduce the rate of staling considerably compared to storing at room temp.

With these measures, the beer should not oxidize much before it's been consumed.
Why would you need to restart the siphon several times? With an auto-siphon, you should have one or two strokes of the wand, and then it siphons until finished.

Also, why are you doing a transfer to secondary at all?

As for air in the line during the transfer, I find if I pinch, rotate and/or constrict the flow near that point where a bubble is present, you can usually "fill it in", and get the bubble to clear.