Ok, but can you suggest a line of moves? I asking about what moves to make and the thinking behind them?
How would you handle this endgame? Also, king question.

By my answer, I would think you have an idea of what to do. White's connected pawns overwhelm the c6 pawn, so the idea is to play c5 preparing a d5 pawn break so white can further improve his position.
It's weird I don't see it as a "break". Do you mean sac a pawn to play c6?

Ok, I see the confusion now. It's not the pawns, it's the rook on the 8th rank!!!
You can't avoid a back rank mate. When I was playing I didn't realize that the rook would be blocked.

General plan: play c5 to immobilize black's doubled pawns, centralize the king, create a kingside passed pawn, use that passed pawn to distract the black rook(s), pick up the scattered weak black pawns. Should black allow it exchange one or both rooks as the kingside passed pawn will give you a winning K+P ending.

After 31. c5 Black has the ressource 31. ... Rb1 (32. Rxb1 Kxe6) forcing the exchange of one rook pair. This would defend against the breakthrough with d5 and make a back rank mate unlikely. Still I think that the pawn majority on the king side gives White the better chances.
How would you play as white in the following position? Also, subsequent moves include moving the king. How would you suggest playing the king?