In a nutshell

CivFanatics has a pretty well written guide for beginners. But that's long.

So here's my condensed new player guide for Civ4:

Tile Improvements:
1) Never work an unimproved tile
2) Food is the most important resource
3) Never work an unimproved tile
4) About 1.5 workers per city early game
5) Overlap is fine; every city only needs a few good tiles, when are you going to get to Pop 20 anyways?
6) No seriously, go Slavery and just whip the guy working that bare grass.

War:
1) Your army should be 50% siege units
2) If you settle a hill city near an AI that might war, it'll be pretty unassailable with archers then longbows.
3) It could happen at anytime

Economy:
1) Cottages on all the grass you can support, Emancipation + Free Speech + Emancipation + Free Market
2) Alt: Farms + Workshops + Windmills + Watermills + Specialists, Representation + Caste + State Property; try to build Pyramids
3) Alt 2: Espionage

City Improvements:
1) Granaries, Forges, Aqueducts mandatory
2) If you don't have anything useful to build, switch the city to Wealth or Research
3) That Bank you're building in that hill city with 5 gold per turn? That falls under Not Useful.
4) If you don't know whether you need that Wonder, you don't