KK in SB multiway on A high flop

[LowDown22]

These hands are always tough for me to play against loose-passives out of position because I never know where I'm at.

Party Poker 1/2 Hold'em (9 handed) converter

Preflop: Hero is SB with K[img]/images/graemlins/club.gif[/img], K[img]/images/graemlins/diamond.gif[/img].
<font color="#666666">4 folds</font>, MP3 calls, CO calls, Button calls, <font color="#CC3333">Hero raises</font>, <font color="#666666">1 fold</font>, MP3 calls, CO calls, Button calls.

Flop: (9 SB) 4[img]/images/graemlins/club.gif[/img], A[img]/images/graemlins/diamond.gif[/img], 3[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(4 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 calls, CO calls, Button folds.

Turn: (6 BB) 4[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 calls, CO calls.

River: (9 BB) 8[img]/images/graemlins/spade.gif[/img] <font color="#0000FF">(3 players)</font>
<font color="#CC3333">Hero bets</font>, MP3 folds, <font color="#CC3333">CO raises</font>, Hero calls.

Final Pot: 13 BB

What should be the typical plan here?
Bet all streets but fold to a raise (which I think I should have done in this case)?