Converting a pawn advantage (or not!)

This was an online game I just finished (1 move per 24 hrs).  In the mid game my opponent gave me a pawn to open a line against my king, and I took it.  I was able to generate some threats on his king though so he couldn't make use of his open file.  It reduced to a rook and pawn endgame with me still a pawn up.  I used a dubious pawn sacrifice to try to break through, and both lost the pawn and gave my opponent the advantage.  However, I used the threat of a passed pawn to clean up his two passed pawns and draw.

How could I have won this rook and pawn endgame?  Or are there any other opportunities I missed?

Thanks for the analysis, but those are mostly variations for Black.  I played White in this game.  How could White have broken through?