I build 3 warriors (or scouts) to go explore, hopefully find a couple of techs or maybe even a settler. Then a Granary, because even though i fall behind the ai civs for a while, i think long-term it's worth it. Then it depends on shields and food. I try to get the city to produce 5 or more shields per turn (ten would be wonderful, since everything is in multiples of ten, but quite hard to do) even after splitting off a settler or worker. [Nine is the worst - look at all you're wasting!] A super food location will mean fewer turns between "growth units" (worker or settler), but usually i can squeeze a military unit or improvement between them.
Note that Barracks is after those first 3 units, but ALWAYS before all other military units, unless barbs come knocking.
The second city depends on its food/shields, as above. But one of the first two must be your settler factory, or you'll fall behind your neighbors in growth, and never catch up. (depending on Difficulty Level.) And as you send out settlers, you need to roughly match with mil units, or barbs will eat them. (on higher barb settings.)
[Is it me, or do some C3C civs really push aggressively close right away, walking settlers right up into your land, and grabbing it before you can? I don't remember this being as bad before Conquests...]
As for research, all i do is keep it the highest setting that still gives me a slight gold income. So over the early game, i'm slowly building a treasury that can help make deals and a few useful embassies. (If the powerhouse of the game is your neighbor, an embassy and RoP and frequent deals can make a big useful friend out of your potential doom.) The exception would be playing an expansionist civ that hits many goody hut tech jackpots; then turn research down and rake in cash until the goody hut rush is over.
A note about early trading: when trading the first techs, ask for whatever cash they have. It won't be more than ten, but they're usually happy to throw it in, and every little bit helps in the beginning.
One more note about the beginning: i research the tech i DON'T need, usually the bottom religious row. This is because the instant goody hut freebies will NEVER be what you're researching, so with a little luck, you can get the more valuable tech quicker and for free. When this works, it's a nice boon, and you need these techs anyway. But at some point, you give up and go research or trade for what you want. Finding where the horses and iron are is important.
