Does a star fuse helium to beryllium on the main sequence? When a star has finished fusing all its hydrogen into helium, it will then start fusing helium into beryllium and so on and so forth up until iron.  

When the star is fusing to beryllium, will the star still be in the main sequence phase and will it at that point start to grow into the red giant phase, or is there no given rule for when it will start growing?
Does a star fuse helium to beryllium on the main sequence?


Stars don't fuse helium to beryllium except as a very, very short intermediate step toward carbon. Helium-helium fusion to form beryllium is endothermic: It consumes energy. To make matters worse, the beryllium-8 that results has an extremely short half-life, less than $10^{-16}$ seconds. Helium would be the end of fusion in stars (and there would be no us) if not for a fluke: The beryllium-8 formed by helium-helium fusion has almost exactly the same energy as does an excited state of carbon-12.

This greatly increases the probability of a third helium-4 nucleus combining with a short-lived beryllium-8 nucleus to form carbon-12. This is stable. The next stage after hydrogen burning is thus triple helium burning (the triple alpha process), essentially bypassing beryllium except as an intermediary.


  When the star is fusing to beryllium, will the star still be in the main sequence phase and will it at that point start to grow into the red giant phase, or is there no given rule for when it will start growing?


A star leaves the main sequence well before it starts fusing helium. It leaves the main sequence when the star can no longer sustain hydrogen fusion in the core. This happens when the core becomes void of hydrogen. At this point, the helium left behind by hydrogen fusion is essentially ash. Hydrogen fusion proceeds on the edge of the core (shell burning), but the hydrogen-depleted core at this point is far too cold to fuse helium to carbon (not beryllium). So it collapses, and gradually becomes hotter.

The star starts fusing helium to carbon (and also oxygen) if the post main sequence star's mass is large enough. At this point, the red giant collapses and behaves almost like a main sequence star with a second life. That second life doesn't last very long, however.