Use either white or yellow onions, but preferably not red ones, as they are less suitable for the long slow cooking.
At
stage 4 of the recipe, there are 2 conflicting schools of thought: some prefer the more traditional method of cooking the onions until they are dark brown, others (like me) only brown them lightly - it's a matter of taste, of course. I remember as a child staying with my grandmother Jeanne. She used to cook her onions until they were really dark and, back then, I loved that.
For the grated cheese, choose whatever you prefer that browns well, such as Comté, Gruyère or even Emmental.