As a huge fan of the game that has loads of problems of time I will just mention the stuff I noticed myself as others have already given general tips.
I am a fan of the east roman faction (byzantines), the thing is that they happen to be one of the most difficult factions to play with enemies in all directions, close proximity to mongols and lack of hard counters to knights. I have a large contempt for the venicians who have nothing better to do but send armies in my direction with better spearmen, heavy crossbowmen and knights.
1. Send diplomats in all directions, try to make alliances with everyone possible. Your reputation gets ruined by prolonged warfare, but this is countered with stable alliances. You do not want to end up in a situation where everyone hates you, it is a lot easier if you can focus on one faction at a time. If you are catholic then an alliance with the Papal State is invaluable, due to increased popularity and the ability to call for crusades.
2. Merchants have zero upkeep and generate money, the result is fairly obvious. Check around the map for high value trade resources. After blitzing through Anatolia and capturing Antioch I realized that I had monopoly of all resources in Asia Minor and Egypt, that equals thousands of florens. Go for the trade nodes that no one is using, no need to manage and a garanteed boost to your economy.
3. Mercenaries, costs a lot but are often helpful. For the byzantines they are vital until you get access to proper infantry and heavy cavalry. Recruit them when you are fairly certain of a battle or seige. Use them to carry the rams and ladders during the seige and feel free to put them in the first line during battles. After major victories you should seriously consider disbanding them, 4-5 mercenary units have quite the upkeep.
4. Militia should be used to bolster the numbers and a lot times that is enough to turn the odds big time. Massing up a full stack of units and going on a raiding spree through a neighbour's country right after they declared war upon you will grant you loads of money and a peace treaty. Also helps when you need to occupy foreign lands as you don't have to spare any professional troops as garrisons.
5. Regarding battle tactics, learn the basics and keep them in mind at all times. Ranged attacks negate melee defense skill, attacks from the rear negate shields and armour piercing attacks halves all armour. This makes mounted archers and crossbowmen extremely lethal if you use them right. A basic cavalry archer with and ranged attack value of 6 will kill anything not in plate armour if you shoot them in the rear (even the pesky Dismounted Feudal Knights). Crossbows makes pikemen and halberdiers pretty useless imo as they will just massacre them (Which makes the Armoured Sergeants one of the most reliable infantry units in the game).
A note about the "Hold" command, it will cause the unit to hold formation regardless of events. Use this with spearmen or cost-effective swordsmen to recreate romanesque field tactics. Heavy spearmen in hold formation is a pretty darn good way to stopping heavy cavalry. Likewise an army with large amounts of Byzantine Infantry makes for a pretty reliable force all around. Nice to have in the important cities.
6. On army management consider veterancy. Keeping the units alive and retraining them will raise their experience, upgrade their armour and they will do very well in battle. Also consider cost-effectivity and versatility. Cost-efficiency combined with large quantities is what wins wars. Norse Archers is a unit that with upgraded armour is the equivalent of Norse Swordsmen, while still having a pretty decent bow. Most mid-range cavalry archers can be very flexible and difficult to counter at times. Militia Crossbowmen is a unit that is dirt cheap but with a fairly decent ranged armour piercing attack. Seriously consider using Mounted Sergeants or equivalent to support your knights, costs a fraction of the nobles while still being able to do a lot while flanking (in my first game ever my Mounted Sergeants got more kills than equivalent number of Mailed Knights).
A final note on settlements, with small towns it is usually far more profitable in the long run to have low taxes as it increases population growth.
