There are 6 cores in total. The first 2 are available through story missions. In general, it is best to leave the remaining 4 until after you complete the story. One of the cauldrons will require the breathing apparatus from a story mission.
I made videos showing the full walkthroughs, minus most of the fights. Hopefully these videos help. There is no audio commentary, but I did chop them up into chapters and added overlays when explanations are needed.
Repair Bay Tau:
Cauldron Gemini:
Cauldron Iota:
Cauldron Mu:
Cauldron Chi:
Cauldron Kappa: