I’ll answer this twice, once generally and once specifically.
Generally, for me, a flop like that depends on position. If I am out-of-position, I will very likely check that to show weakness and induce a bet. If Villain bets, I CALL, do not raise or you will scare him off. Then you can repeat this on the later streets and cha-ching! if he keeps betting. On the river you can then check-raise instead of check-call. If he slows down on the turn then I have to start betting value. If I check the flop and he checks back, I need to start betting some value on the turn. We tried to trap and it didn’t work, oh well. Sucks but whatever. We do the best we can.
If I am in position and Villain checks, then I want to put out a nice easy value bet. If he folds here then we weren’t going to get anything anyways, so no harm done. On the later streets we can escalate nice and easy, think something like 20% flop, 25% turn, 35% river, and you can vary that up according to the blinds/stacks, vibe of the Villain etc. I would probably not check back because you are wasting a street of value here. Sometimes you might be up against a Villain that you know well and you know that he loves to bluff and get aggro when you’re weak. In that case you can check back the flop and he will likely get aggro on the turn, but that’s a special case. If I am in position and Villain bets, I am flat calling 100%. Make him feel confident of continuing. I would probably even tank before I call to fake a tough decision. ; ) I would probably only come overtop on the river.
Still generally speaking, I am never checking more than one street because I’m throwing away streets of value. Don’t try to outsmart someone by playing possum for 2 or 3 streets. If they are not willing to offer you money, you’re going to have to start asking for it sooner rather than later. If they fold to small sizing, we weren’t getting paid anyways, so don’t lose any sleep.
Now, specifically, you are in 3-way here, 2nd seat of 3. BB checks. This is more complicated but probably better for us. Now normally we want less Villains but flopping quads we smoke everyone so we don’t care if we are 9-handed. Best case scenario we’re up against AA and set-of-rags and they both stack off. But more likely, ONE of them is willing to call our value. With 2 Villains we have double the chances of getting a value call. : D So I would put out a standard non-scary value bet here and see who bites. Next streets, continue from there, escalate nice and normal. If the pot is nice and big by the river we could even jam. Watch their stacks. If the pot ends up bigger than their stacks they might have to call anything (pot-committed).
But the main piece of wisdom here that I can offer is to not be afraid to bet sensible value. If they fold to small sizing you weren’t going to get anything on later streets anyways so you lose nothing. Obviously don’t bet large and scare them off, but yea.
You do also mention letting them hit and improve. This is a good point, but generally I just want my value. We can’t rely on the board to help us. The board has helped us enough! : D Furthermore, specifically in this hand, there is no straight here. You didn’t mention the suits, if there is no flush draw here then all that Villain can improve with is 2-pair, set, boat (and you block a lot of boats here), so they are not likely to improve much. Just go get your value.
Would love to hear from the more experienced players on this.