Some extreme basics:
You control your demesne. This is your personal holdings, and you only get so many even if you’re the grand high mega-Emperor. Economic action only takes place on the Barony or County level.
Counties are individual map bits; baronies are the individual squares in them that represent cities, monasteries, or castles. Each County also has it’s “top-level” holidng that represents the County itself, which is almost always a castle, but can be a city in rare cases - you see this around Italy especially.
In most cases, you want to control as many holdings as possible before hitting your demesne limit. If you look in the top-right portion, you’ll see a number like 3/5 or 7/8 or something. This says how many holdings you have, and how many you can posess without hitting nasty economic penalties. Always aim to control as many as possible unless you really know what you’re doing.
On the intrigue screen, you can unleash plots. These are good for getting yourself up to the county maximum, because you almost always have access to a plot if it would net you a holding instantly. (I have no idea how the hell the game designs plots, as it’s something of a sore point for me that I can’t create my own at will). It’s easy to use - just pick the plot and start adding people to the party. You want to get your evil schemes over 100% in most cases.
If you intend to keep control over a holding, it can be a good idea to invest in making it more profitable. You can invest in each holding to add structures, and each structure gives you more money, defensivness, soldiers, or other special bonuses. The money-gaining ones are a really good investment until it starts costing 200 - 300, and then they’re worth it for the long haul only. As you invest in the cash some more, you should also start buying troop-producing structures to get more soldiers.
When it comes to warfare, there’s some odd bits to understand. You get Levies and Garrison troops. Levies are dudes you can roll around and kill with. Garrison troops just sit there until they get beseiged down. The more Garrison troops, the longer it will take for an enemy to defeat you - defenses like walls and such affect this as well. In order to even start beseiging, you need to outnumber the garrison, plus any levies still sitting at home.
Usually, wars follow several patterns: once war is declared, both side try to raise as many soldiers as possible, then the AI goes completely bonkers and makes as many possible Epic Fails as it can. It’s especially bad at handling big stacks of units in hostile territory.
In any case, what you want to do is concentrate your forces ASAP, and hit the enemy with everything possible. Try to run around smacking little units and avoiding their big troop concentrations - attack when you have them outnumbered as least two-to-one if possible, and hire mercenaries if you can afford them to wear down the enemy without really costing you any soldiers, then disband the mercs after slaughtering the enemy armies. Then settle in and seige anything vulnerable or which is the object of your war effort. Take the slow way and seige targets instead of atacking unless you have an overwhelming advantage (like 10-to-1). Depending on the war, the AI will usually surrender at between 80%-100%.
Beware of things like Holy Wars - they sound great and can net you huge territorial gains, but the AI can and will rope in shocking numbers of allies. I think I once had to wipe out about three times my numbers in better-tech’d enemy soldiers, and only managed it through the use of Holy Orders, calling up every single soldier I could lay my paws on, keeping them active dangerously long, and chewing the enemy up peicemeal. That said, remember that you don’t have to win the war if it’s going against you. Just surviving it often enough.