the sun - Why do stars become red giants?

The destiny of a star basically depends upon its mass.
All its activities variety depends upon its mass.
If a star's core has a mass that is below the Chandraseckhar limit ($Msim1.4M_{sun}$), then is destined to die as a white dwarf (or, actually, as a black dwarf in the end).
The composition of the white dwarf, also depends upon the original mass of the star. Different masses will lead to different compositions.
More precisely, the more massive is the star, the heavier are the elements composing the final object.
This is because more mass means more gravitational potential energy

$dU = - frac{GM(r)dm}{r}$

that in turns can be converted into heat.

The hydrogen nuclear fusion starts, for the proton-proton reaction(that is the dominant process for Sun-like stars) at around $10^7 K$. This is the value that allows the particles to overcome their coulombian barrier (i.e., to fuse).
After the hydrogen fusion, when the most of the core is composed by helium, then of course the hydrogen fusion can't happen anymore. The core starts to collapse, and heats itself. For a Sun-like star, there is enough mass to compress up to a level that heats the core enough to start the He burning. But that is all. When also the Helium is converted into Carbon, the star has not enough mass to compress again up to a level that starts another nuclear fusion reaction. This is why the core nuclear reactions stop. For the shell burning question, it is important to understand two things: $(1)$ the shell structure of a star is only an approximation, and $(2)$ there is a gradient of temperature within Sun-like stars, that means that (besides the corona) the temperature increases when you go from the outside to the core. Now, if the core is compressed and became so hot to burn helium, the shell "outside" the core (that in a onion-like schema was within the radius of the previous hydrogen burning core), is still hot enough to burn hydrogen. The size of the helium-burning core is smaller than the hydrogen-burning core (this is compression by definition). The shell has still enough hydrogen, and contemporary is deep enough inside the star (that means high temperature), to allow nuclear fusion of hydrogen.
If the star was more massive, more things could happen, like heavier elements core fusion, and more and more burning shells.

Take a look at these:
Ref 1, Ref 2.

Ref 3 for some numbers too.