What was I supposed to do?

I should have won this endgame, but I played it terribly. What idea was I supposed to follow to make this a win?

I think you were doing fine except for 29 .. Rd5 giving up the back rank mate.  You want to play with your king in the endgame.  I would have brought him up to f8 and e7 instead of just making luft with a move like h6.

yeah Rd5 was wrong - actually he could've played Rb8 picking off both of your rooks one at a time for mate

Hi Citadel_7--you lost because of basic tactics at the end there, but you're right about this being a victory for black otherwise--endgames with two extra pawns are much easier than those with one extra pawn.  Starting at move 25, you hit on the right idea of getting your rook behind your passer with either ...Rc7 or ...Rc8.  You floundered a bit on the next few moves; there was no rush to shove your pawn and ...a6 was a mistake since that pawn is better defended on a7.  King safety was key, taking a turn for ...h6 or similar, and that would've left you with plenty of time to maneuver your rooks around your c-pawn.  Think about the final position: if your pawn were back on a7 it'd still be alive, your king would have a way out since you could've used that turn for breathing room, and you'd be ready to deliver a crushing blow with ...Rd1.  So definitely some right ideas here, but also some neglected defense in crucial parts of the board.

Thanks for the advice everyone. I pushed the pawn to a7 because I got worried about his passed pawn pushing home on me. I see now that my king needed to get more involved in the action.