I recently played a game that ended with this endgame position. White couldn't seem to put enough pressure on any weak spot with his two rooks to win a pawn, and as long as Black kept moving his king and rook back and forth to protect the e and g pawn, there were no real threats. If someone could try and analyze from this position, it is right after black sacs his knight with white to move.
I'll post the final drawing position too.
On the original position you showed, one thing to notice is that your king is stuck defending the e6 and g7 base pawns. If your g pawn was to come to g4, there is only 1 pawn the king has to defend. Also your rook should manoeuvre around the board to behind your h6 pawn in order to push it up the board. These two ideas might be your long term plan. In my opinion it seems like a delicate situation for both players, if black pushes everything up the board everything may get stretched thin and white can take advantage of it quickly with both his rooks, and if white gives black too much freedom, black can immediately take the advantage and get on the move. If black to move a line might go something like: 1...Ra2, 2.Rb8 ...Ra7, 3. Rh4, and hes stopped your play, if the g6 pawn moves the h7 pawn hangs, if you play 3 ...Kg5 after 4. Rh1 ...g6 5. Re8 you have to defend the pawn with your rook and it becomes very passive. If instead of 2...Ra7, 2...g5 was played, he doesn't have the option of Rh4. He might play 3. Rh8 ...Kg7, 4. Rcc8 ...g4, 5. Rcg8+ ...Kf7, 6. Rf8+ ...Kg6, 7. Rfg8+ ...Kh5. Here he might play 8. Ke3 ...e5, create a barrier, 9. Re8 ...f4+, 10. Ke4 ...Re2+, 11. Kd3 His king is back where it started...f3, If his king moves you move the f pawn up, if his h8 rook moves you move king to h4, any other move such as Re6, respond with Kh4. e4+ would be a game losing move because after Kd4, g3, Rhxh6+, Kg5, Ke5 and black cant do anything to stop Reg6# mate (this is still mate even if black plays f2). g3 would also be losing because after Rhxh6+, Kg5, Reg6+, Kf4, Rh4+, Kf5, Rxg3, any variation to that black ends up losing a pawn and will surely lose. After 12. ...Kh4, 13. Rhxh6+ ...Kg3, whites got nothing he can do, black's next turn is e4+ kicking whites king off the square in order to be able to queen the f pawn without the rook being taken, you can also do this with Re1 but you gain a tempo with e4+.
Any questions? (if i stuffed up any notation coordinates tell me)