No stats on villain. Bet fold turn? Played horribly?

Full Tilt No-Limit Hold'em, $0.10 BB (5 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

Hero (Button) ($19.90)
SB ($8.13)
BB ($19.57)
UTG ($11.01)
MP ($10.48)

Preflop: Hero is Button with 10, 6
2 folds, Hero bets $0.30, 1 fold, BB raises to $1, Hero calls $0.70

Flop: ($2.05) 4, 10, 2 (2 players)
BB bets $2, Hero raises to $4, BB calls $2

Turn: ($10.05) 6 (2 players)
BB checks, Hero checks

River: ($10.05) A (2 players)
BB bets $14.57 (All-In), Hero folds

Total pot: $10.05 | Rake: $0.67

Just fold pre, you're getting good odds but your hand is terrible, i'd call flop and fold to much furthur action, hes unlikely to barrel the turn as a bluff or with worse for value.

Flop raise I'm not sure I approve of, but if you are going to raise, raise more. Once the turn comes I want every excuse to get out of the hand.

Yeah fold pre definitely. As played, fold river, he doesn't bluff that soaking wet board

I'm assuming that if you're opening this on the button the blinds are either bad or very tight - in either case its a fold unless you are a postflop master and can run circles around this villain.

I hate the raise on the flop. If raising was any good, it would have to be at least $8, but I wouldn't bother because he'll continue with a high club or shove over you with an overpair anyways imo. If you think he's cbetting his whole 3bet range on that flop you can call and bet/fold a checked turn (club or not) as a bluff or to protect your hand. The villain would have to be sick or holding a made flush to barrel that turn after a call.

So, depending on his cbet tendencies, call, then b/f turn, b/f called river.

If he doesn't cbet much, just fold flop.

I would't bet the turn. I also agree with iIImatic about raising the flop. If you're going to raise there, why the min-raise? I would either commit there to a larger raise or call off the flop and give up to any further action. On second thought, I might just give up on the flop as it seems pretty weak to just call and give up when there could be a lot of bad turn and river cards. What are you doing if an ace or king rolls off on the turn instead of a 6?

Hero is Button with 10, 6 2 folds, Hero bets $0.30, 1 fold, BB raises to $1, Hero calls $0.70

Found your leak.

Fold preflop.