Hi!
During the first few thousand years, depending on my situation, I generally follow this pattern:
Build city.
Build defensive unit (usually militia).
Build settlers (if there is a food surplus of 2 or more).
Build temple if city still has a decent food surplus, OR granary if it doesn't, OR city walls right away if i know I'm sharing a big continent.
If temple was built, build granary; if granary was built, build temple; if city walls were built, build barracks (since I'll likely be at war soon).
After a temple and granary have been built (in a "safer" city), build a marketplace, OR another settlers if: A) its first settler has built a new city or; B) it still has a BIG food surplus.
If city walls and barracks were built, then I usually go crazy with chariots and attack like mad until I have the whole continent, or I have created a LARGE distance between my civilization and my nearest neighbor's. THEN I build a settler, and then I start building temples, granaries, etc.
By the time my cities all have temples, granaries, and marketplaces, and most of them are building libraries, I generally have Trade, and seem to be working on Religion around this time (I go for Religion as soon as possible!). Thus after a library is built, or in some cases right after the marketplace is built, I build caravans and cathedrals. Usually I will designate a couple of high-production cities to build caravans (and a few ships) for everybody, while the rest build cathedrals, and often I will build the two religious Wonders as well (I like them).
At this point, it is usually around 1ad. Now I am sending ships laden with caravans and diplomats to other continents, my cities all have cathedrals (except for any newer cities, of course), and I am ready to REALLY play Civ! For me, 1ad-1500ad is the most fun period: I spend it sending out boatload after boatload of caravans and diplomats, exploring my opponents' civilizations, and building universities, banks, and city walls at home -- until I develop Gunpowder (usually around 500ad), when I spend a couple centuries building barracks, musketeers and cannons.
I only play on King Level, by the way.