How do i play the anti-scicillian?

That's not the anti-Sicilian. It's just the Sicilian.

I think the best response for black would be 3..d5.

/I don't know what you mean by saying the black should have pushed to f3. Black can't push to f3. Perhaps you mean that black should play his pawn to f6?

You just have to mirror every move from a sicilian. You can sugest your opponent to start first, even tho you're playing with the white pieces.

Ok, so first of all, if the word "defense" is in the name, it's something black chooses to play, and if the word "attack" is in the name, it's white.  Often shortened to "the Sicilian," but it's also called "the Sicilian defense," meaning it's named after something black chooses to do: (1...c5 against 1.e4).

Therefore an anti-Sicilian, is something white does.

And so when you ask how to play against an anti-Sicilian, you're asking for tips for black!

Anyway, I'll assume you're asking about white.  There are many different variations of the Sicilian, but often the first few moves by white and black are:

Now about anti-Sicilians, there may be a better definition, but basically an anti-sicilian means you're not letting the game get this kind of pawn structure:

Anything that doesn't allow this structure (after the moves 1.e4 c5) could be called an anti sicilian, although there are a few time tested ways to do it, and when people say "anti-Sicilian" they're usually referring to 1 of a few different tries by white.

Here are some: http://www.chess.com/blog/Torkil/anti-sicilians

Now about your move.  Pushing e5 is a good move if you can keep the pawn on e5.  However this early in the opening it's too far away from its army.  Black can play d6 at some point and basically force white to capture meaning 2.e5 just loses time.  Also, defending it with f4 is a bit passive (blocks the c1 bishop) and weakening (loosens the kingside)... again only because the rest of the army is still not in the game.  When all your pieces are developed, e5 supported by f4 can be great (because now your developed pieces can use the space).  After 3.f4 black can consider (like notmtwain said) the move d5.

As far as anti-Sicilians go, I prefer the Rossolimo, where white is playing 3.Bb5 and later will play c3 to support d4 or c3 d3, and so you get pawn structures resembling a closed ruy.