QQ, rivers full house on AK high board. Bet size?

Only been at the table a few hands - no history on villain.
Pre-flop is standard I think
On the flop I've been altering my c-bet size a lot recently, too small here?
Turn I don't want to fire again.
River, ok. Hand ranges. Villain min-raise/called pre-flop. I think range is likely AQ/AJ/TT/JJ. AQ will call any raise most likely, the rest fold to a big bet. Thinking about it now I like one of $10 or push. I believe that some hands will call $10 but fold to a push, especially as I look as though I'm slowplaying AK/KK after the action pre. Thoughts?

Full Tilt No-Limit Hold'em, $0.50 BB (9 handed) Full-Tilt Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver Cards)

CO ($47.10)
Button ($45.95)
SB ($47.25)
BB ($49.75)
UTG ($99.55)
UTG+1 ($10)
MP1 ($20)
MP2 ($41.45)
Hero ($49.25)

Preflop: Hero is MP3 with http://www.flopturnriver.com/phpBB2/.../smiles/qd.gif, http://www.flopturnriver.com/phpBB2/.../smiles/qc.gif.
UTG raises to $1, 3 folds, Hero raises to $2.5, 4 folds, UTG calls $1.50.

Flop: ($5.75) http://www.flopturnriver.com/phpBB2/.../smiles/4h.gif, http://www.flopturnriver.com/phpBB2/.../smiles/kd.gif, http://www.flopturnriver.com/phpBB2/.../smiles/4s.gif (2 players)
UTG checks, Hero bets $2, UTG calls $2.

Turn: ($9.75) http://www.flopturnriver.com/phpBB2/.../smiles/ah.gif (2 players)
UTG checks, Hero checks.

River: ($9.75) http://www.flopturnriver.com/phpBB2/.../smiles/qh.gif (2 players)
UTG bets $4, Hero raises to $?

I make it 14 to go

First, your continuation bets should be somewhere between 1/2 of the pot and the full pot. Any less and you're not denying pot odds on all sorts of draws, regardless of whether you actually have a hand or not, and any more and your risk-reward ratio starts to suck.

Second, the pot is $13.75 to you, so a raise to somewhere around $13-15 seems about right.

Your bet sizing on every street is too small. Make it at least $3.50 preflop and bet 3/4 on the flop. You should consider betting the turn, an A is often a good card to double barrel and will fold out mid pocketpairs most of the time (which I think is his most likely holding here).

I raise to $20 because I think a random A and possibly a K calls.