Fold bottom set on scary board/turn? -- .25/.5 NL FR

[Ichoi]

This is full tilt, 9 handed .25/.5 NL. No numbers, my only read is that he's a fairly average player, haven't seen him get out of line in the 20 minutes I've been playing.

I have

UTG+1 Raises to $2, I call right behind him, everyone else folds.

Effective stacks $52

Pot [$4.75]

Flop:
he checks, I bet $4, he calls.

Turn: [Pot 12.75]
he checks, I bet $10, he min raises.

[DoctorHandles]

[adamh]

[Danhdan]

This is one of the ugliest boards for a set I've seen in a while. That minraise is scary and he has a big hand, although he could easily have A here. I call the turn. I am still not sure what to do after I call and ___ hits the river though. A 4th heart makes this an easy fold if he bets, I'm also not wild about any Q, J, or paired board unless it's the . I'm probably trying to keep the pot small on this one...no raise even if the board pairs. Wow, that is weak tight...maybe if a ten comes I feel better about calling a push.

9 or 10. A blank doesn't seem too wonderful either, but I'm still calling a push on the river if the blank hits.

[Stoneburg]

He has a bigger set.

[Danhdan]

[Ichoi]

At the time I was really thinking he had KK, and by the small size of his raise, probably with the heart. No other hand made sense to me.. maybe but that felt like a long shot.

But being a relatively inexperienced NL player I decided "ah screw it, it's only .25/.5 he might be getting stupid on me" and shoved.

He had KK with the and I immediately wish I just went with my read and folded.

edit: River was the