Principle for converting in the endgame

I feel like I should have won the game below if I knew how to capitalize on the pawn advantage at the end.  After 25 . . .RxD3, which I thought should be winning for me, I felt like I was just drowning in choices, and made all the wrong ones.  By what principles should one try to exploit that kind of ending, or was I truly lost?

That is the particular question I have on this.  Any analysis of any part of the game is welcome, of course.  I am new to trying to post PGN files, so pardon me for posting the whole game.

I know I'm not a very good player, but this type of endgame is a key weakness, especially under time pressure.  With enough time, I can usually just calculate my way in with brute force, given an advantage, but that is not the same as being aware of the principles in play.

Thanks anyone who takes a look at this.

by below, in the first sentence, I meant above.

You allowed him to get a strong passed pawn 34 ... h5?

you missed opportunities to make your own passed pawn, eg 40 ... dxc4?

19...Rxg3 mate (useful theme of bishop pinning f2 or f7 pawn)

You recognize this endgame should be better for black, good!  So play confidently in this knowledge.  Play 31...Rxg3+ and go into the winning endgame without those doubled kingside pawns.

Similary with this confidence you can play 34...dxe (also he won't get a passed pawn this way).

34...h5 seriously weakens your pawn structure.  g5 is a hole and g6 is a target.  Don't expect a passed pawn to push past an enemy king and rook all by itself.  You can only push a passer home if get all your pieces involved.

Similarly 36...c5 is weakening / a waste of time.  What you really want (as is common in rook endgames) is have your king take over blockading duties for the rook.  If you only learn 1 thing from this game it should be that leaving your rook passive like that can only ever be bad.

So Kf8, Re8, and Ke7 is the natural sequence here, switching out blockaders.  The rook's activity is > the king's is the logic behind it.

Absolutely the only move was 43...Rxe6.  You played 43...h4?  You again think this pawn can queen on its own?  There's no way.  It's exactly the same as not developing pieces in the opening (actually due to few pieces on the board it's worse :)  Every move you didn't play Rxe6 was not good.

To answer the way the topic was posted, the principal in rook and pawn endgames is activity.  Top three priorities in order are #1 Rook activity, #2 king activity, and #3 pawns.  It's often you see a master sacrifice a pawn here or there in a rook endgame to maintain activity of his king and rook.  Passivity is the kiss of death.

Not that you can throw away pawns all over the place.  Just be sure to jump at the chance to get your king and rook mobile.  Rooks hate blockading (limits their mobility arguably more than any other piece).  Rooks like to be on the enemy's 2nd rank (usually where the weak pawns are) and rooks like to be behind passed pawn (their own and the enemy's!).  An active king is in general a centralized king.  It can also be the side of the board where there's a passed pawn (yours or the enemy's).

As far as pawns, in endgames the more secure pawns are the ones closest to home.  The more they move forward the more vulnerable they become.  Not that you shouldn't push your pawns, but do so carefully and only if your non-pawns can support them.  Your game would have been much easier if you had first worked on mobalizing your king and rook, and only then moved your pawns.