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I'll just make some random observations - hope they'll be useful. I'm a beginner myself.
The effective stacks for this hand are 196.5bb.
Even UTG I would probably raise this pocket holding. At the level you are playing (both in terms of the stakes, the skill of opponents and your own skill) I would play 77 mainly for set value.
I hit a set about 1 in 8 times, so I should not commit more than in the best case a tenth of the effective stack before the flop - in the normal case I should be looking to have the effective stacks be 20-25 times more than what I put in before the flop. This basic observation suggests that any raise before the flop up to 8bb ($0.16) is gravy for my hand.
Before the flop I could decide to $0.10 expecting noone to re-raise, or I could raise to $0.08 allowing an opportunist to min-raise me to $0.16 and still have my entire pre-flop play be +EV in view of me playing for set value.
So yeah: Raise PF.
As played, the flop comes low connected and rainbow. Everyone who is in the hand has limped in, which allows anyone in the hand to have hit the flop - two pair hands, open ended straight draws and paired and unpaired overcards are all possible.
At this point I need to start betting my set for value. I also need to make sure that straight draws do not get a good price to improve to a better hand.
I would take advantage of the fact that noone on the flop respects the pot size (and bets relative to the pot size) and are probably almost as likely to call a $0.20 bet as they are to call a $0.10 bet - if they have a hand they'd like to play a little they will, if they don't, they'll fold regardless. I would probably bet somewhere between $0.15 and $0.20. I would tend to favour $0.20 because $0.16 looks so very considered where $0.20 can look just lazy. $0.10 and $0.12 bets are also ok in my book, but $0.08 is needlessly small.
On a drawy flop it is generally wise to bet a little higher to ensure that people who are on draws don't get a good price for their draws. If the pot had been raised pre-flop people in it would be more likely to have high cards and less likely to have low cards, and based on that you could consider the available draws on this flop unlikely and thus cut back a little on your bet size.
Flop: Betting is ok, but I'd have bet more.
Turn: Good card here - if someone has decided to call the flop and stay in the pot with high cards that could mean they have a king. They could just have improved to a second-best hand that they are willing to pay you a lot of money for.
I could see myself betting out here instead of going for a check-raise. If the king didn't improve his hand and he doesn't have two pair or a straight draw he'll probably fold, but he'd fold anyway to a bet on the river, so you don't lose any money that way. If he has a king and is willing to pay for his king you need to start to get him paying sooner, so the bet on the river that he'll call can be bigger. If he has a straight draw you need to continue to charge him for it - similar to the flush draw now on board (though it's somewhat less likely).
When you decide to check-raise - I really don't like the min-raise. When you min-raise you are giving him great odds to draw (in this case to a flush, but it could as well have been a straight). If you do decide to check-raise and he bets about a third of the pot I would consider it likely that he's sitting on a drawing hand where he'd like to put a little extra money in the pot at a good price to sweeten the pot for him on the river if he does hit his hand. If I was check raising here I would put in a healthy check-raise. A pot sized check raise would be raising to $0.10 x 3 + $0.28 = $0.58. I would probably raise to something like $0.50.
Turn: Lead out or check-raise bigger imo.
When the river comes that's a pretty terrible card for you. I've been paranoid about the straight draw all along, and while people holding a 6 would have had an open ended straight draw and just made a pair now, the betting hasn't really effectively been pricing out even people with gutshots - so a villain could still be sitting with 44 or 99 (or K9) giving him a straight. Being a diamond it also completes the flush.
While initially I thought I could agree with betting on the river but would just bet smaller, on second thought I'm not sure I even want to bet small. I have a hand with showdown value, but a lot of draws that were not effectively priced out just completed. While a set has good absolute hand strength in this situation it has pretty mediocre relative hand strength. If I bet small (like half pot size) I'd still expect to be called by two pair hands and overpairs (and kings) and not expect them to raise me that much. If I get raised on my bet - either the half pot one or the one you made - I'd have to consider folding. I'm not saying I'd fold, but I'd have to seriously consider if I'm maybe beat. I'd still end up calling this bet, but I wouldn't expect to win half the time.
River: I'd bet smaller (or check/call), but still call the raise as played.
In summary I think the problems in this hand are lack of aggression on behalf of the hero pre-flop, flop and turn (when your hand is probably the best) and too much aggression on the river (when your hand is probably the second-best).
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