Opponent was new to the table so no read

Party Poker No-Limit Hold'em, $0.5 BB (9 handed)

MP3 ($50.7)
CO ($47.4)
Button ($19.4)
SB ($50.85)
Hero ($64.7)
UTG ($50)
UTG+1 ($46.07)
MP1 ($6.5)
MP2 ($41.3)

Preflop: Hero is BB with , .
3 folds, MP2 raises to $1, 2 folds, Button calls $1, SB calls $0.75, Hero calls $0.50.

Flop: ($4) , , (4 players)
SB checks, Hero checks, MP2 bets $3.8, Button folds, SB folds, Hero raises to $10, MP2 calls $6.20.

Turn: ($24) (2 players)
Hero bets $15, MP2 calls $15.

River: ($54) (2 players)
Hero checks, MP2 checks.

Final Pot: $54

Results in white below:
Hero has As Kd (one pair, kings).
MP2 has Ad Ah (one pair, aces).
Outcome: MP2 wins $54.

Lead out the flop and fold to a raise or check raise like I did? Should I go into check fold mode after his flop call? I should be lucky he didn't bet the river.

Before I even finished reading this post it was completely obvious that he had AA. Obviously, in the moment it may not be so easy but if you go back and look at the play it makes sense. First of all the $3.80 bet, I'm assuming this was the exact size of the pot post-rake which means either that he checked the bet-pot button or he typed in the exact pot-size either of which roughly equate to eachother. The bet pot-button is scared agression usually. The guy knows he's got a good hand, but he's not all together happy about the flop he got and realizes that there's a lot of ways he could get beat if the hand goes on. Then you go back to the pre-flop action, the min-raise, it's nice to know before hand if this is aces or not, but given the lack of a read, you've got to at least entertain the possibility. Now he flat calls a sizeable check-raise, it is OVER, if a king doesn't hit the turn it's check/fold. I think after the flop action the only hand he could ever hold here would be AA, QQ is an outside possibility but in sum, yes this hand is over after that raised gets called on the flop for me.

I think it's better to check-raise than to bet and fold. If he has AK or KQ; he will raise your bet on the flop and you'll fold a winner/split. If you check-raise and get called you can assume that he has at the very least TPTK.