What I like to do to advance in science: (this has worked for me on Chieftain, Warlord, and Regent) Huge map/ Pangea/ 16 civs
1. Expand fast, to start I usually build spearman, settler, spearman, settler, spearman, etc. (varies by city). I just keep producing military units when my cities are down to size 1, then when it looks like I will complete a settler after I will reach size 3, I'll produce the settler, then go back to military units. The military units will just be used to fortify my city (2 can be used in each to keep the citizens content) with the extra units being sent out to wait for a settler (scouting out the area), or to defend a newly built city.
If you have an expansionists civ build a few scouts and set them on 'auto-explore' (just hit the 'e' key). They will find you alot of techs from goody huts on a large pangea map with no barbarians. I have played some games on cheiftain where my scouts did ALL of my ancient tech researching.
2. After your capital has produced two or three settlers, you have two options.
a. Build wonders, letting your other cities become the settler factories (this might slow your expansion down a little), especially if your capital is built on a river (free aqueduct) , with good production squares.
b. Give up on building the first few wonders (all but the Pyramids become obsolete later on, anyways)
3. Once I have claimed as much land as possible (I start running into my neighbor's territory) and I have 2 spearman in each city, then I switch to producing improvements. Preferred order (if they are available to be built, and depending on what I need for happiness, cash, etc.) :
Aqueducts and Courthouses varies by city.
Marketplace (commerce and happiness bonus)
If you have a city up in the mountains, or in the desert, and it is limited to a size 2, then some of these may not be needed, so you can save some money on upkeep costs.
I don't build barracks (because I will get Sun-tzu's), don't bother with walls, either, since once the city reaches size 7 they are obsolete. I don't bother with graneries, either, since my cities seem to grow fast enough without them.
I just build all the other improvements that are possible (courthouses aren't needed in your capital, though). I set my science to 50 %. Luxury rate at 0 % until alot of your cities start reaching size 6, then up it to 10 or 20 %. When you run out of things to build, build workers (I like to have an average of 1 worker/city. If a city has everything it can build and I already have enough workers, then I will build a barracks and start producing military units for a possible invasion of my neighbor to expand my territory. Or I will have the city produce Wealth. Use the whip on some of the improvements that will take way too long to build in your extremely corrupt cities.
At the very end of the ancient era is where the AI seems to stop trading techs with each other so often, so try to buy a tech from the civ that is the only one with that tech, then sell it to all the other civs for techs you don't have, and/or for gold/turn. Once I get out of the ancient era (don't bother with Monarchy, get Republic) don't sell any techs at all until I know I'm 3 or 4 techs ahead of everyone else (the AI really slows down it's research rate in the middle ages, especially if you bribe them with a tech to go to war with another civ). Every civ will join an alliance with you for a tech, plus they will pay you gold/turn!! (make sure you ask for the gold), and you don't fight at all. Declare war, get your allies, but don't send any units out. This enables me to get a huge lead on all the other civs on the techs and production of the Great Wonders (have a city building the 'palace', that you can switch production from once you have the tech for that wonder).
In your early expansion phase you should have been able to get 15-20 cities or more , and with all those improvements and with those workers making sure you have roads everywhere, you should be able to get techs every 4-6 turns with your science rate around 50-60 %, halfway through the middle ages, in Republic, then later Democracy.