Is a Pilgrimage a Quest?
In short, no, Pilgrimages are not Quests. They are to be taken alone. When the Pilgrimage is undertaken, and successful, you acquire access automatically to either the Blessed Mark or the Cursed Mark depending on which route you took.
Should you acquire the Blessed Mark, you may also unlock any of the additional upgrades offered by the Guardian Spirit. You will need to choose (1) one of these additional upgrades after completing your Pilgrimage.
Once a month has passed, your character may travel to visit the Guardian Spirit and acquire a second upgrade. You are NOT required to take the pilgrimage in the same way again, you need only travel to visit The Guardian Spirit as your previous Pilgrimage has unlocked access to them freely. A single submission showing their communion will unlock a second upgrade.
Should you desire a third upgrade, simply travel again a month later.
Your character may commune with the Guardian Spirit whenever they please once they are Blessed by them, but must do so only when the Guardian Spirit is known to be active.
What if I have already been Blessed by a Guardian Spirit but wish to take a Pilgrimage to another?
In this case, you would follow the necessary steps for your chosen Guardian Spirit's Pilgrimage and once successfully completed they are now similarly unlocked for you to access as mentioned above. This will apply to all Guardian Spirits you take Pilgrimages to.