You played this just fine after the flop.
You definitely want to raise more PF, though, as this will probably get hands like Q2 suited to fold. The min-raise does nothing except build the pot. Think of raising to at least 4xBB, to get the trash cards out of the flop.
I agree with your bet on the flop - a pot sized bet, because you really prefer to take the pot at the flop rather than see an A or K on the river.
Your opponent goes all-in. Unless you really believe he just hit a set or 2 pair, you have to call. You're most likely facing a hand that thinks it has top pair with good kicker (like AJ, KJ). Against the top pair w/ kicker, you're winning about 80% of the time.
If he has the two pair or trips, well, you don't get the odds
to call really.
I've played this level of poker before - and I don't consider it to be poker. Too many people care nothing about the money they are playing with OR with increasing their bankroll - all they are playing for is the thrill, and so they make wild and wacky plays constantly. I wouldn't call your opponent's all-in a bad play if he had the A, but Q-high flush draw isn't an ideal place to semi-bluff. Of course, there are people who do this even on $5/$10 poker, but not so many, and they lose their money a lot faster.
I think you read his hand correctly (or at least valued your hand correctly), figured you were ahead and called. He got lucky on you - but thats part of the game.