Preflop I'd bet more, probably 1.25 (I play at party, which is looser, 1.00 might be a good raise at stars).

I'd raise the BB's flop minbet to 2.00 and fold to any reraise.

As played past the flop, I'd raise the min bet to 3/4 of the pot and fold to a reraise.

Notice how I keep saying "and I'd fold to a reraise?" If you would have played this hand the way I am saying, you'd have found out where you stood on the flop and saved yourself a stack.

Now you were stacked by a set, but what I think happens here more often is that you get stacked by two pair. You can't bet positive that the A helps you on the turn in a three way pot where the people behind cold called your preflop raise. AQ or AJ could have sucked out on the flop and paired their kicker on the turn.

I think the 10 supposed outs (3 A's, 3 K's, and 4 T's) you had on the flop we're enough to warrant a raise of the bb's minbet. 9 times out of 10 you take down a decent pot right then and there. In this case, you'd have likely been reraised by Mr. Set and been able to get out of the hand BEFORE the Aces started peeling off the deck.