Is the sound you hear a groaning noise or a squealing noise? If it is a squeal then your belt is shot and you can replace it yourself with a manual. If it is a groaning noise it is the hydraulics or you are just low on fluid (not likely it it goes away when it is warm.)
You could also have a worn out power steering rack. Mine has very similar symptoms and my PS is pretty much gone when the rack is cold.
I think it is a combination of worn hydraulics and the cold shrinking all the parts. It is about 10-15 degrees at night where I'm at. That makes all the parts on the car shrink just a little bit. I do have a worn rack but it is being magnified by the "shrinkage."
I think that shrinkage is making the seals on the rack leak a hair more, resulting in a loss of power steering. Once my car is nice and warm the problem is all but gone.
So if you are having the same problems it is probably the rack too. Go under the car and see if it is covered in fluid and goo. If it is, it's probably shot.