What should I have done instead

In looking back over this hand, I definitly feel I played it wrong. Thus I turn to the experts at FTR for advice.

Party Poker No-Limit Hold'em, $10NL (9 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

UTG+1 ($7.60)
MP1 ($3)
MP2 ($19.94)
MP3 ($4.32)
CO ($8.52)
Button ($11.39)
SB ($5.80)
BB ($10.20)
Myke ($10.50)

Preflop: Myke is UTG with T, T. SB posts a blind of $0.05.
Myke raises to $0.35, UTG+1 calls $0.35, 2 folds, MP3 calls $0.35, 4 folds.

Flop: ($1.20) K, J, 3 (3 players)
Myke checks, UTG+1 checks, MP3 bets $0.1, Myke calls $0.10, UTG+1 calls $0.10.

Here I figured I had the pot odds to see the turn.

Turn: ($1.50) T (3 players)
Myke calls $10.05 (All-In), UTG+1 calls $7.15 (All-In), MP3 folds.

Here I was worried about a flush draw, but after thinking it over I think the push was a definite mistake.

River: ($18.70) 3 (2 players, 2 all-in)

Final Pot: $18.70

Results in white below:
Myke has Ts Td (full house, tens full of threes).
UTG+1 has Ah Qd (straight, ace high).
Outcome: Myke wins $18.70.

The results looks like more luck than skill, but I did have 10 outs to the Full House or Quads.

I am no expert. Here is how I would have played it though.

I would have either limped or raised to .4-.5 preflop. People don't respect preflop raises very much at these stakes. .35 is ok, but I think .4-.5 might have worked a little better.

On the flop I would have checked and raised villains weak bet (probably to like .9). Check/calling here isn't a bad option, I just like to raise people who bet weak, no matter what my cards are. Usually you get a fold (sometimes people will bet weak with a monster so you will do this, learn to spot these people and write down notes).

On the turn, I would have bet the size of the pot, with the intention of calling a push reraise. Even against a straight, I have a 25% chance to win. And its usually not a straight that pushes over the top. This board makes a lot of two pairs.

If just called I would obviously value bet my boat on the river for like 2/3 of the pot.

No expert here either. I agree with Renton on limping or raising more. On the flop I would've definitely raised big to like $1 or so just because I have this part of me that wants to kill people who min-bet in these situations. Oh ya and also cuz you don't want to let him price his draw - which I find min-bets to be lots of times. If you get re-raised take note of it and remember for next time. I probably wouldn't open push the turn like that cuz you lose value doing that I think. Bet normal and call the push.

Wouldn't betting that much money on the flop be a bad move since with 2 overcards I'm like not to have the best hand, and then drawing to only two outs?

You would not betting the flop per se, you'd be attacking his apparent weakness. If he reraises you can easily fold and note that he's probably minbetting good hands to induce bluff. You could do this with any two, but in this hands you have the added bonus of possibly having the best hand right now, you have your two outs that are probably good, and you have some backdoor draw possibilities as well.

As played, I'll bet out the turn and call his push.

Remember to note that he minbets his draws in position instead of taking a free card.

Oh, and don't go AI if you worry about someone being on a flush draw. You want him to call incorrectly if he is, so bet out the largest amount you think he would call.