I have just completed Chapter 3 of my class story. I have been sent to Ilum and if I remember rightly Ilum is just a daily hub. Does this story advance on Ilum? or should I just go back to my ship to begin the next expac?
It's not only a daily hub although the story is short and the quests become dailies after the first time you do them. It goes Ilum quests>Battle for Ilum (flashpoint)>False Emperor(also flashpoint).
That being said you can safely skip it and solo it at 55/60, finding groups for lvl 50 flashpoints can be pretty hard.