You'll start acquiring the power of the Six through the main story, from Chapter 4 on. The requirements for summoning them differ, but you shall find them appearing should you struggle on your way. Once you fit the condition, you'll see a notice that you will need to hold
to bring the Astral forth.
TitanAvailable starting: Chapter 5
Condition: Your comrades have fallen. This increases the likelihood of him appearing, but you also need to make sure you're in a large enough space for the Archaen to manifest. Best chances for this would be stressful fights out in the field.
RamuhAvailable starting: Chapter 5, halfway through
Condition: Lengthy fight duration. The longer the fight takes, the more likely Ramuh will manifest with levyn lights and lightning surrounding. The Fulgurian is impatient, so the longer the fight, the better.
LeviathanAvailable starting: Chapter 10
Condition: Danger status for Noct. After you unlock the serpent, the chance of spawning it forth in battle comes from the longer you are in a weakened state. This might be a bit too risky as your health will plummet through retaining this form.
ShivaAvailable starting: Chapter 12
Condition: See all three above, combined: fallen comrades, danger state, and a long battle.