Two moves is the answer, and Black is the one who gets it. After 1.f3 e5 2.g4 the black queen swings to h4 and the game is over before White has touched a piece.
8 
7 
2 
a 
b 
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d 
e f g h 1 There is no shorter mate and there cannot be. One move each is not enough to open the diagonal to e1 and land a queen on h4 at the same time, whichever moves you choose. This particular disaster has a name, Fool’s mate, and it is the standard first illustration of what checkmate means.
Three moves for White, and why the extra one
White moves first, which sounds like an advantage and here is not. After White’s second move Black has only played once, so the pawns on f7 and g7 are still standing and the black king cannot yet be got at. White needs a third move, and the position looks like this.
8 7 2 a b c d e f g h 1 Same recipe with the colours swapped. The f-pawn opens the diagonal, the g-pawn charges past the square that could have blocked it, and the queen lands on the far end. There is more on the three-move mate if you want the alternative move orders.
What actually happens in real games
Not this. I have played in tournaments for over twenty years and the shortest game I have ever finished was eleven moves, and it ended with my opponent resigning a piece down, not with a mate on the board. Strong players resign; they do not sit still for two moves while you set up a queen check.
Short decisive games do happen, and when they do the cause is almost always an opening trap in a real variation, or a blunder that drops a piece and then a second one that drops the game. That is a different skill from memorising Fool’s mate, and it is the one worth training. If your games keep ending early on the wrong side, the fix is in how to stop blundering rather than in any move sequence.
So what should you play for?
Development. Every fast mate on this page comes from the loser pushing the f- and g-pawns in the first two moves, and no opening move you make can force that. Two-move and three-move mates are worth knowing so that nobody catches you, and worth nothing as a plan.
If you want speed, get it from the middlegame instead. A tactic that mates in two out of a normal position is a real skill, and two-move mating puzzles will teach you more in an hour than the opening lottery ever will.