Played a hand badly?

I had K10 suited, played in late position. Limped in last, guy checks, and bets a tiny min bet in this freeroll on FT on a 10QQ flop, no flush draw. I'm the only caller, and my plan is to just call tiny bets, and see what happens.. (bad plan??) Turn is a Q, and he bets that min raise again (we both have about average stacks). I call. River is an A (I'm like uh-oh, I played this wrong). He does the same min raise (he thinks I have nothing maybe?). I call, and he had J9 suited....

What should I have done?

Any paired flop should raise caution flags. You hit 2nd pair, and I would think if your villain either slowplay checks, or raises a decent amount, he hit his Q. Not the case. He is first to act and often a min bet can get the field to fold. He should have known that your call suggested you were traping or hit the 2nd pair. From his POV his bet is a blocking bet. He creates enough unsureness in your mind that you don't make a raise and instead just tag along. Doesn't tell him a lot does it. But it did prevent you from making a big bet, which if you were first to act, you should probably make, most of the time.

The 3rd Q on the turn, has freaked him out as much as it freaks you out.
Again his blocking bet prevents you from betting your boat. But it shouldn't have. In this particular case, Its very likely that villain would muck to any pressure, and with a brick (for him as well as you) river A, he is hoping to scare you cheaply.

It may be that you made as much as you were gonna be able to make with this hand.

So I would suggest you take a dif approach. Games eat time, and time is money. So when you hit the 2nd pair flop, raise. Ignore his blocking bet, and raise. He mucks, you win and you all get on to the next hand.