I'm really new to the godot engine but I think the scenes are already a FSM.
Each scene represents a state (scene menu, scene game) and you can go from one scene to an other (one state to an other) with
In your case in the menu screen just connect the new game button signal
pressed to a method that will load the game scene.
If you really want a FSM just go with an autoload script (Singleton) keeping the current state of your game with a basic state pattern.