1. Make sure your council is always hard at work. What they do is really up to you but there's two things you ALWAYS want to be doing. First, tell your spymaster to uncover plots in your capital. You have a base chance to always discover plots whether your spymaster is actively counter spying or not, but doing this boosts that chance significantly. If you discover a plot in its infancy and the character who is the mastermind likes you enough, you have a chance to resolve the situation peacefully. If you dont discover it in time it usually leads to a civil war.
Secondly, always, ALWAYS tell your court chaplain to build relations in Rome. Excommunication is the single most powerful tool you can use against your fellow catholics, but you cannot get excommunicated if the Pope likes you. Avoiding getting excommed should be near the top of your priority list
2. Wars are the best way to lose useful characters - never think that this game is like EU3 where characters hardly ever die when in battle. Sieges are bloody, and if you attack a province dealing with an outbreak of a disease it can quickly turn into an absolute mess. This is a bit gamey, but in my opinion you should always tell your king to stay at home (unless you're caught up in roleplaying and want to be the warrior king on the front lines)
Raise your crown authority to anything above Low, that way you can manually assign the leaders of your armies. Give command of the army to people who are either low on the totem pole (courtiers, barons, ect) or Dukes who hate you. You shouldn't have any shortage of people who have at least decent martial stats, so don't worry about that. What you cannot risk is losing your supporters even if you're fighting for your life against a foreign invader. There's no point in winning a war if you lose your most trusted advisers - this is how civil wars start!
3. Always keep an eye on who your vassals are allied with in the allies tab. The natural reaction when seeing your vassal is allied to a lot of your other vassals might be to think nothing of it. Actually, this is the worst possible situation you can ever be in. A vassal who has built a strong web of inner-realm alliances is a far bigger threat than any blob on the map. If he rebels, he's going to bring all those allies against you (unless they like you more).
Inner realm politics is like a constant cold war. You want to limit the personal power of your vassals while maintaining a strong base from which to fight. To that end you should always keep a dimense size of at least 50% your limit if not more. Even if everyone in your realm likes you, all it takes is a bad succession to find yourself fighting for your son's survival against men who his father thought as his best friends. Opinions are fickle, a strong personal army last forever.
It's also a VERY bad idea to give any Duke more than one duchy. From time to time they will get the "desires Duchy X" ambition for themselves, which serves as a negative to their opinion to you. My first natural inclination was to just give them what they wanted, and while this will satisfy them for about a decade, its not long before they start getting too big for their boots and they try to take on the crown for independence. Remember, its all about balance.
4. Keep an eye on the culture of the tutors your children get sent to. Children who grow up with an Irish duke, for example, will take on his culture. Lets say you're playing as France and your wife is Irish so you let her tutor them. Sounds good right? Unfortunately, your children will take on her culture and when they take over your realm, your vassals will hate them because they view them as a foreigner. Always, always
send your children to nobles who have the correct culture to be tutored.
On that same note: when choosing a tutor for a child think about what you want your child to become. The card furthest left in a characters traits will tell you what kind of person that character is. You'll see things like "brilliant strategist" "midas touched" ect. Personally, I like to send heirs to get tutored by those with high bonuses to Stewardship and Diplomacy, this will give you a boost to income and relations respectively. I tend to send the second or third sons to get better martial training as I like them leading my armies. With any luck they will die in glorious battle and not even think about contesting the throne when their brother takes over
5. When you take over a new province sometimes all of the vassals who used to serve there get thrown out. Unless you want to control every single holding in a province (making your demesne unmanagably large) you'll need to assign new vassals to those holdings. Dont worry, the game actually rewards you for this since you get more income from cities if a mayor rules them than if a feudal lord (you) does. To create new vassals, bring up the province menu and right click on one of the buildings there. If you can make a new vassal, the "create new vassal" button will not be greyed out.
6. Finally, dont forget to check the "include all lower titles" checkbox when you're handing out landed titles. This will ensure that when you give out a county, all the vassals of that county will go with it. If I want my brother to control county X, then naturally, i want him to be the lord of all the vassals there. If i forget to click that, then while he'll have the title of "count of X", i'll still control most of the vassals who live there. In short this simply allows more power to the people who hold titles in your lands. Some people dont do this and actually limit the power of those beneath them.
Honestly i've seen good arguments on both sides, but I tend toward stronger servants. It makes management less of a headache.