How can I play this hand better?

This is my first post, and I have to tell you that these forums have been very useful. This is going on memory so it may not be entirely perfect. Now to a recent hand:

Blinds 20/40 $5 SNG at UB.
Hero 1400
Villain in EP 900

Hero in MP with AQo

Villain calls 40
Hero calls 40

Flop comes Q 4 7 rainbow

Villain checks
Hero bets 100
Villain calls

Turn is 5

Villain checks
Hero bets 175
Villain calls

River is 3
Villain bets 300
Hero pushes all in
Villain calls

You can probably guess what the Villain had. So I know I played the river horribly. My question is, should I have bet more on the flop or turn to make him pay for drawing? I have run into these circumstances a number of times and haven't really ever picked up the right way to play them. Any suggestions are fine.

Thanks,

Han