I would play Bd6 first, because it's more flexible.
First, it allows you to castle more quickly. Second, you know that you want your bishop on d6, but you don't know exactly where you want your light-squared bishop. Perhaps it belongs on g4, pinning the knight. Third, if you do want to put your bishop on f5, it might be worthwhile to first prepare this with h6, so that you can meet Nh4 with Bh7. The pawn formation on f7, g7, f6 and g6 didn't work out too well for you in the game, as your king didn't really have a good square; it was vulnerable on g8 to the light-squared bishop, it would have been vulnerable on h8 to checks along the h-file and it would have been vulnerable on f8 to a check on h8.
I think f5 is a fine square for the bishop, and if the knight moves to f3 then g4 would also be a good square. I personally would prefer g4 in that case since it seems more active to me; it pins the knight whereas a bishop on f5 has no target. But I don't think it's a mistake to eventually place the bishop on f5. It's just that the timing is important. You might have to prepare it with h6 to meet the threat of Nh4. The one square I don't like putting the bishop on is e6, since after Bxe6 fxe6, the e6 pawn becomes a weakness. It can easily be attacked along the e-file, and by a queen on b3 or g4. And if you ever play e5, then either you end up with an isolated pawn on e5 (after dxe5 fxe5) or you end up with pawns on dark squares, which is the wrong color to place your pawns when you have a dark-squared bishop.
I think the real question is not whether you play Bd6 or Bf5 first, but how you meet the threat of Nh4 when you have a bishop on f5. That seems to me the reason you lost the game. You had no good way to meet this threat and that gave white a lot of initiative.