A similar hand happened to me a few days ago and am wondering how this should be played.

You are dealt Qs 7s in BB. 3 other people limp PF.
Flop comes Ks 6h 8s. 2 people check and 1 bets, you raise hoping to get a free card on the turn. The 2 people fold and the bettor reraises.

Your play?

Um If you're in the BB how do you raise to get a free card on the turn? This makes no sense.

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“There's no sense in being precise when you don't even know what you're talking about.” - John von Neumann

I am sorry, you checkraised. Does this make sense, I am still new to limit?

No it doesnt make sense. You can't get a free card when you are first to act.

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“There's no sense in being precise when you don't even know what you're talking about.” - John von Neumann

Check-raising for a free card ain't the best play, but you call now, and check-call the turn.

If you got in this awkward position, after he 3 bets you call. Check call the turn and check fold the river if you dont hit your flush or backdoor straight.

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“There's no sense in being precise when you don't even know what you're talking about.” - John von Neumann

How would you have played this from the beginning?

Straightforward. Check-call the flop
check-call a missed turn, but out if you make it.
check-fold a missed river, bet out if you made it.

The reason you don't go for a check-raise on the turn like everyone else does, is it gives your hand away. The stop and go makes it look like you are stealing and you are likely to gain extra bets if he raises.

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“There's no sense in being precise when you don't even know what you're talking about.” - John von Neumann

Just call his bet on the flop.

I would bet the turn if you hit your flush, but it kind of depends on what you know about the player. Check/raising might be the best move actually, now that I think about it. Then, bet the river.

If you don't hit you flush on the turn, I might just fold since you haven't lost that much on the hand. But it depends on if you hit a card that gives you a straight draw. Then, I would call his turn bet.

EDIT: Check/raising the turn might be advantageous, because he is likely to call the raise. You will get a little more out of him. On the river, you bet again, and if he folds, it's all right. But he will probably call again. I don't know if it would give his hand away, because you have know idea what these guys play. He might have two pair, and will call any raise. Or, he could be the guy who can't let go of TPTK.

I agree with this 100%, no point in getting fancy here.