Chess.com had a wonderful Game of the Week a few weeks back, which showcased the promise of Hypermodern play. I tried it in this game, but got all tied up. I could really use some help...

I think you should've played 21 gf.  Once you allowed his knight(s) to get hold of e5, your position was suspect.  And after all, you did manage to get a draw out of that endgame (so things could be worse).

One of many themes in the Reti is preventing your opponent from completely taking the center. You are playing as white, so (unlike in the Pirc) you can play in hypermodern style by allowing d5, but preventing e5.

In your game, you strayed from the Reti and went into the KIA. After "1. Nf3 Nf6", play "2. c4" in order to grab the remaining central square. You still had a chance to stay with the Reti Gambit if you played "3. c4" rather than "3. Bg2".

In hypermodern openings, you do not advance the center pawns and, therefore, do not open lines of attacks to your king. You really do not need to worry about forming the fianchetto and castling as quickly as possible (otherwise you will end up playing a reverse KID). Just try to fianchetto on the kingside within the first 10-12 moves.

I am a very, very casual player, but still know a fair amount of the recurring themes in various hypermodern openings. Just ask anything you would like to know about the Reti. I might be able to recommend some lines based on your personal preferences (single or double fianchetto, specific set-ups against black's defensive options, transpositional possibilities, etc.).