You already finished the game.
First, my evaluation of the position: Black is a piece up (a knight). Add to this the superior placement of black's pieces as follows: black's queen is MUCH better positioned, black's dark squared bishop is also MUCH better positioned than its counterpart. The light-squared bishops are about equal, black's knight is I guess well positioned for defense (important but let's face it it's not positioned for attack). The rooks are a slight advantage for white b/c of the Rg1. All of this adds up to a probable winning advantage for black, even without the extra knight.
So I think you've played quite well to achieve this position.
In terms of how the position may continue - what I'd do if I were to move, as black, is probably play axb4, and then ... c3 (pawn sac) to liberate the e6 bishop, and then play ... Bc4. This exchanges a LOT of pieces, and leaves black with a winning position a piece up, and possible some mating possibilities, depending on who exchanges rooks.
I'm Black and White is about to move. My question concerns his pawn attack on the Kingside. I'm sitting there with three pawns on their starting squares and he keeps advancing. Should I have begun to move these pawns earlier rather than concentrating on moving my pieces to positions that would give me a winning game?
Thanks.