The original edition was here.
3 years since we finished chatting about it and actually started! Anyway, we didn't develop rules for advancing mysteries. The best solution for this is to suggest the rules if you were the alpha SG for a saga, and outline what you believe is sensible and won't wreck your beloved saga. Everyone else can make suggestions based on their experience, and we can develop something that others can use as inspiration.
Mystery cults probably should be mysterious with their progression being uncertain and requiring strange tasks from their initiates.
For Holy cults - it's probably easier to either use the Pilgrimage mechanic from The Church, suggest what you're trying to achieve and then have the other contributors suggest how many seasons they think it takes based on your current skills;
or to use contemplative mysticism (good news - in the Levant joining a Crusade is very easy. Bad news - you'll really need to work hard at not being found out by Quaesitors).
Actually, using contemplative mysticism is probably quite easy if your character is really pious. Use the Major personality flaw Pious, and have a +6 Pious personality trait. Use the points gained from the Major flaw to buy the Major Virtue of a holy Method at character creation, and buy a holy Power.
Further Minor holy powers and supernatural abilities (like sense holy/unholy, premonitions) require a Path total of 15 (The Church, page 22). Your Path Target is your Path Total - score in Personality Trait Pious - so you only need 9 points.
9 points can be gained from one year on a Crusade; or a season preaching to heretics, pagans or infidels; or one season teaching and spreading the gospel (+3) and undertaking a minor pilgrimage (+6) - all of these are easily found within the Levant, it's the not getting killed doing it that's the hard part. Alternatively, nine seasons of serving the poor or in personal devotion to a saint could do the trick.