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I agree with the above posts that the biggest problem was in the J7 hand, not the 88 one. You said you're a beginner, which means that you shouldn't be playing in marginal situations with marginal hands. J7 certainly is one of those hands. It was also rather silly to assume that an UTG+1 minraise wouldn't have you dominated, unless you had a read that he'd raise / minraise every hand or some crazy stuff. The standard read here is that he has a monster. Minraising in early position is kind of like limping UTG. So yeah ... AA, KK, QQ, JJ, AJ, KJ, QJ, JT, etc. all dominate you.
Now that doesn't mean that its an automatic fold. Personally, I usually call minraises on the BB if I have a decent stack, looking for implied odds (such as hitting a 77 or J7 flop when he has an overpair or TPTK or whatever). On this specific flop I personally probably would've just called on the flop and gotten my read strengthened based on what they did on the turn (where I then typically have enough info to have a strong read on their hand). If he checks the turn and its not overs, you usually would bet it, as he's respecting your call on the flop and is apparently scared of the top pair. If he bets strong, you usually have to fold. If its a medium to small bet, you need those more detailed and player specific reading skills.
Anyway yeah, it gets real complicated with hands like this, and as a beginner you should look to avoid these type of situations and just fold your BB (and possibly look to make most of your blind defense plays preflop, where things are less complicated, and you won't have to worry about playing against better post flop players that have better hands, which is a big time losing situation).
After that, I almost certainly would've went bust with the 88. With 10xBB left and a decent position, I don't see myself not shoving all in. As has been said the 3xBB bet and then call of the all in is kind of silly, but it really doesn't make too much of a difference when you do call. Folding after raising so much of your stack, and with a decent hand too, would've been really really weak. Limping would be better than raising 3xBB and considering folding, but its still pretty damn passive, and you can't afford to be that passive as a small stack. So yeah, shove instead to make them know they're going to have to play a big pot to call you, and hope not to get unlucky and run into a monster like AA next time!
One last little thing that may help a beginner player that is feeling bad about going out like that. Once you're short stacked, its kind of best to kind of lose all fear of busting, ready to accept that fate at any time. That doesn't mean you should necessarily be all in every hand, but there's no real reason to be afraid or feel bad about busting out at this point (especially with any real decent hand like 88). You're going to need some luck to double up, and there's no way around that. The big stack really has more to fear, as they have all the potential and so many more chips to lose to mistakes. Having the opportunity to play as a short stack, instead of busting straight out as a big / medium stack can be considered as a kind of freeroll to get back into the mix of tournament if you can double up once or twice.
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