Opening

The only moves to avoid chessmate are pawn at d5 or f6.What is the right move in this situation?

Usually, I would play Nh6.

There are more ways. In the opening, you'll want to develop pieces towards the center. Look for a move that develops a minor piece towards the center. Try and handle their threats by making a move you want to make anyway, not a move that you really rather don't want to make.

Free tip: 2. ...e5 is not a good move. You've already contested the center with 1. ...c5. It's better to try and get your pieces developed and not spend time on a second pawn move. Once you'll learn about outposts, you'll also realize another reason why the 2nd pawn move is not a good idea, but that's for later. Usually black tries to prepare the move Nf6 with a move like 2. ...d6 (otherwise white can kick the black knight away with pawn to e5). 

This is a game which I lost it and now I want to understand all the opportunities which I had

A very good move.Another move can be qf6 to force a queen trade

3...d5 loses a pawn. 3...f6 weakens all white squares. 3...Nh6 4 d4 renews the threat to f7 with 5 Bxh6. Most logical 3...Nf6.

Which is very good and why I basically gave you the hint to find the move on your own. #6 beat you to the punch though.

Don't play e5 after c5 on move number 2. If you play the Sicilian you will certainly not play 2. ...e5. If you wish to play e5, do it on the first move and don't play Sicilian.

White's move 2.Qf3 is not great as it develops his queen too early, plus it takes away a natural square for his knight. It is sometimes a sign that the opponent will play Bc4 trying the Scholar's mate. You can play e6 as a second move, making Bc4 ineffective and at the same time you are preparing to develop your bishop. Other options are Nc6, just developing a piece, or d6 that prepares Nf6.

White's second move (2. Qf3) was bad, but Black's second move (2. ... e5) was even worse.

Better was 2. ... e6 which takes all the sting out of White's Qf3 / Bc4 idea.