tl;dr: The best time to form your own religion is immediately after you just fought in a successful crusade for your religion, because not only does that give you plenty of Piety, it also drastically reduces the cost of forming a new heresy.

Farming Piety

Fight battles against non-believers

Each battle against an army controlled by a character of a different faith awards both prestige and piety.

Fight in crusades, and win

Winning a crusade pays out a large amount of Piety from the "War Chest" depending on each participants war contributions. If you manage to max out the cap of 20%, then that can be a pretty huge lump of piety.

Pick the "Learning" lifestyle.

Not only does increasing your Learning stat increase your monthly piety growth, this lifestyle also has a lot of random events and decisions with various opportunities to gain some additional piety.

Make use of Piety-increasing decisions

One thing every religion can do is going on pilgrimages.

But depending on the tenets of your religion, there are also other things you might be able to do to earn some extra Piety. Like religions with Communion being able to buy piety for gold by "buying indulgence" from their head of religion, those with Ritual Celebration who gain piety from feasting or Ritual Sacrifice which gives you piety when you execute prisoners.

These opportunities don't get you that much Piety, but every bit counts.

Reducing the cost

Pick up the "Prophet" lifestyle perk.

You can find it under Learning -> Theologian and it halves the cost of creating a religion. With just 3 prerequisite perks, it's pretty easy to pick up.

Wait for your religions "Fervor" to become low.

The cost of forming a new religion is proportional to the Fervor of your current religion. When Fervor is down at 30%, you only pay 30% of the default cost. So this is a very important factor.

Fervor is similar to "Religious Authority" in Crusader Kings 2, but it is a lot more fickle. Various events have a lot more influence on it, but it also doesn't take long for it to revert back to the default. So when something happens which lowers the fervor, you have to act on it relatively fast. Events which reduce fervor are:

• Your religion winning holy wars and great holy wars (crusades). This is a notable difference to CKII, where winning holy wars increased the religious authority of the victorious religions. So the best moment to form a new religion can be right after your religion won a great holy war.
• As you might have also learned from notifications, secrets of temple holders being exposed. So you might want to order your spymasters to "find secrets" in their location so they might discover some dirt on them. Do not expose them immediately. Wait until you are ready to form your religion for maximum effect.

I stumbled upon this entry from the CKIII FAQ posted by Paradox themselves:

Q: I understand that the head of the most powerful house head becomes the head of the dynasty, but what determines who is the most powerful house head? Which factors are the most relevant?

A: Military strength is the main factor

But it appears that when the head of the dynasty dies, then the new head is determined before the titles of the old head get passed on the heirs. As your primary heir will usually not have that many titles and vassals at that point, the head of the house title will often pass on to a different ruler of your dynasty.

Fortunately, that's only temporary. A couple months later, the game will reevaluate the military strength of all dynasty members, see that your character now has all those troops from the inherited titles and vassals, and give the head position to the now strongest character (you).