Is it just me, or is succession the absolute worst thing that could happen to you? Having just conquered most of the British Isles, my old ruler died and the completely rightful heir to the throne naturally became king. But then everyone hated him, a bunch of "X for Skotland" factions popped up, and the Isles basically became just as fragmented as they were before.
Succession is basically the game hitting the reset button. This isn't always bad except for two things: there are now a bunch of pretenders, and instead of your nice old long reign bonus, you now have a short reign penalty, which typically drops your levies by a lot. There are four things I do prepare for my own death and succession:
1. Educate my heir myself, and make him likable. Raise his personal diplomacy, give him virtues, and try to last long enough to have an adult male heir. If possible, try to die with as much prestige and piety as possible as they come with relationship bonuses.
2. Have a large personal army. This means retinues, as many and as upgraded provinces as you can manage, and having money to hire mercs. You want a larger army relative to your vassals.
3. Try to get as close to the optimal vassal setup as possible, which is no vassals with more than a single duchy, and everyone having only one province under their direct control. Not everyone agrees with me on this, but in my opinion once you get to this point rebellions should stay manageable especially as your realm grows.
4 Try to be significantly more powerful than your neighbors. You're going to be having reduced levies as well as internal problems, so you don't want to deal with external threats.
When succession occurs, immediately go into damage control. Use your council members to improve relationships and your spymaster to discourage vassals from schemes. Give important vassals honorary titles and distribute gifts to vassals who have near 0 relationships if you think you can afford it. If your spymaster doesn't like the new ruler fire him. Look through the vassal tab for powerful vassals who hate you and consider extreme action (plotting to kill, assassination, imprisonment). Time is on your side; your goal is to get to the point where most of your vassals like you, and the long reign bonus should basically assure that if you survive.
Not sure how much of that advice was obvious; I can never tell with this game.