burning hydrogen (and maybe other things)

Factual Questions
1

Remembering an old science lesson we were taught that once the sun has used up all its hydrogen it will start burning heavier elements like helium etc. and go red giant in our faces.

But why is it just using up hydrogen at this time?

Surely the nuclear fusion is occuring at the core and since hydrogen is the least dense element it should be “higher up”, for lack of a better term, than helium. But then if the whole sun is “fusing” then there should be a multitude of heavier elements by now?

Can someone give me the straight dope?

2

Put simply, Hydrogen burns easier than Helium does, and so on. Thus, Hydrogen will burn as long as it’s readily available, and when the Hydrogen is used up, the temperature will rise until it’s hot enough to burn Helium into Carbon and Oxygen.

You are right that the Helium sinks to the core. What you get is a Helium core with a Hydrogen-burning shell around it. Eventually, especially in heavier stars, you can get several layers, and several shells of fusing atoms, getting heavier as you go deeper into the star.