This hand was about midway through the session. My question here is: Do you check this turn card? Or was I right leading into it? If I WAS right about leading, was my bet sizing good? Should i bet more or less here?

PokerStars No-Limit Hold'em, $0.02 BB (6 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

SB ($0.37)
BB ($3.25)
Hero (UTG) ($2)
MP ($1.70)
CO ($3.54)
Button ($5.50)

Preflop: Hero is UTG with ,
Hero raises to $0.08, MP calls $0.08, 2 folds, SB calls $0.07, 1 fold

Flop: ($0.26) , , (3 players)
SB checks, Hero bets $0.24, MP calls $0.24, 1 fold

Turn: ($0.74) (2 players)
Hero bets $0.30, MP raises to $1.38 (All-In), Hero folds

Total pot: $1.34 | Rake: $0.05

Results in white below:
MP didn't show
Outcome: MP won $1.29