How do you make your mouse able to move in first person?

You can try this way:

game.Players.PlayerAdded:connect(function(Player)
    wait()
    Player.CameraMinZoomDistance = 2
end)

What this script does is to set the Minimum zoom distance of each new players to 2 (which is the minimum value from which the mouse can move). However, the player's character won't turn to the mouse position in first person view by doing this).