Work on a strategy. Don't automate workers and probably don't send units to move more than 1 turn (unless in your zone of culture).
The computer recommendations are almost never good. You can usually pick a better tech./settlement spot/building/unit, etc. The computer doesn't understand your strategy.
I recommend a granary and a monument in each city (at least monument to get first culture pop for large cross). Plan to work tiles with enough food to max out the fat cross (this should grow as tech advances).
Food
is king. Next is research
and hammers
. Of course you need a good economy
to support your empire so nothing can be neglected.
I'd recommend choosing a VC and trying with a specific civ/leader. Establish an economy best suited to the VC. Have a backup VC.
I'd recommend working on a 'rush' as well. Whether you beeline a tech and go for a world wonder, or you beeline a tech for military advantage and zerg your neighbor.
Build stacks of doom as those are the most effective. By catapults you should always send seige weapons in offensive attacks (upgrade as tech allows).
The AI blows on the water so if you like war and pick a good naval civ (or civ with naval UU), you should be able to win handedly on island or archi. maps.
I have a generic rule about viewing every city in the city screen each turn as well as units moving in the map. I set groups of workers to build or improve tiles so they complete tasks faster. Never automate workers aside from while they are completing the assigned task on the one tile. The rare exception is the build a road to a specific destination (but only with protection or entirely in my cultural border).
Never send settlers solo! Just DON'T!
I try to get at least 1 unit per town and stack to have Medic I upgrade. Medic II is even better so you keep the stacks side-by-side.
Defenders should get Garrison and first strikes, imho. Don't neglect a medic here either. Medic II is great for border cities in case you have to defend from AI. In my experience the AI will attack outside-in, often (note: not always). But I mean they don't really venture into the heart of you civilization for a 2 population filler, if they are marching past a 5 pop with strategic resource.
The rest is usually situational.
Read the Sistuil (sp?) beginner guide.
:edit:
I, too, am returning to Civ 4. It's on my tower in the basement that still runs XP and doesn't use Internet. I also have Dead Space 1 and 2 (maybe 3), GTA 4, The Witcher, and Civ 5 as well.
My first game back and I feel like such a noob. I'm also thinking of switching from marathon to epic just so I can accomplish more games. Not too much time to play at this point.