Endgame technique: How to win with rook + g and h pawn vs rook? The tablebase suggests it's a win for White, but I struggle to beat my phone. Black's rook seems to always be able to harass the white king to stop white make any progress.

What's the technique to win this endgame?

[fen "8/8/6k1/6r1/8/8/6PP/5RK1 w - - 1 61"]
As we can see from the other answers, there are two things you should avoid: 1) The pawns getting blockaded. 2) Your king getting cut off from the pawns.

It is not the only way to win, but as a matter of technique it is a good thing to advance the h-pawn before the g-pawn. In this way your king can use the g-pawn to hide both from checks from the side and checks form behind. But first use the rook to chase the opponent king as far back as possible.

Example:

[FEN "8/8/7k/R7/6PP/6K1/8/1r6 w - - 0 1"]
[Title "Rook endgame g+h"]

1. Ra6+ Kg7 2. h5 Rb3+ (2... Rg1+ 3. Kh4 Rh1+ 4. Kg5) 3. Kh4


The checks will stop and you can think of the next step.

[FEN "8/6k1/R7/7P/6PK/8/8/1r6 w - - 0 1"]
[Title "Rook endgame g+h"]

1. Ra7+ Kg8 (1... Kh6?? 2. g5#)  (1... Kf6 2. g5+ Kf5 3. Rf7+ Ke6 4. Rf6+ Ke7 5. h6) 2. h6 Rb5 3. g5 Rb1 4. Kh5


Here I would again chase the king away before advancing the h-pawn another step.

If the g-pawn is pinned there is no point in giving a check first.

[FEN "8/7k/R7/7P/1r4PK/8/8/8 w - - 0 1"]
[Title "Rook endgame g+h"]

1. h6 (1.Ra7+ Kh6) Rb1 (1... Rb5 2. g5 Rb1 3. Ra7+ Kg8 (3... Kg6 4. Rg7+ Kf5 5. h7))2. Kh5 Rb5+ (2... Rh1+ 3. Kg5) 3. g5


In this way you can advance slowly rank by rank. Towards the end black will have to defend passively to avoid mate at the back rank. Just for the fun of it I would try to end the game with a knight or bishop promotion like in my answer to this question.

[FEN "r6k/8/6PP/5RK1/8/8/8/8 w - - 0 1"]
[Title "Rook endgame g+h"]

1.Rf8+ Rxf8 2.g7+ Kh7 (2...Kg8 3.Kg6) 3.gxf8=N+!(3.gxf8=B!)(3.gxf8=Q??)(3.gxf8=R??)