Alfredo is simple to make -- it's basically cream gravy with a pinch of nutmeg and a bay leaf, then Parmesan cheese added at the end.
If you don't have any Parmesan on hand, do try it with different cheeses. I've used Romano, white and/or yellow Cheddar (for a great mac-n-cheese), Muenster, Gouda, and whatever else I've had on hand. For an even creamier sauce, melt in some cream cheese. And definitely experiment with multiple cheeses. Some of the best cheese sauces I've made were with two or three cheeses.
Every time I've done any of the above, provided I kept the total cheese amount to a reasonable level (somewhere around a cup or so), it always turns out great.
One final note: if you don't have nutmeg or bay leaf, make it anyway. It still tastes good without those.