The things that lead to fast research:
#1: Cities placed to make efficient use of the land around your capital. In the early game, this means cities fairly close together, because each city can only work 6 to 12 tiles tops.
At the very beginning, skimping on other things (say, waiting till later to build that temple) in order to get settlers out sooner will get your economic base going faster, and you WILL make up for the slow research at the beginning.
#2: Roads. Every single square that a city is working, MUST have a road. This is law. It is your Bible. If you see one of your citizens working a square that isn't roaded, you are evil and naughty and should be
'ed.
(Well, leave out the
part, but if you see a citizen working a non-roaded square, get a worker over there as soon as you can. Or better, THREE workers on grass or plains, so you can build the road in a single turn!)
#3: Make an early run at Literature and put libraries in those cities that produce enough research points to actually benefit from a library. (I also tend to go after Republic early, but that can be dicey if your empire isn't ready)
#4: DO NOT build "everything, everywhere"!!! Every building you put up costs money to maintain; if you build too many military units, they cost money too. Eventually, all these costs will force you to start shoving the science slider to the left.
In the Ancient Age, your cities can usually be defended reliably by one or two spearmen; a stack of five or six swordsmen can do a lot of damage to anybody you go to war with. Catapults are a convenience, not a necessity for invasions. You don't need 20 of every single type of military unit. And don't be afraid to disband old units such as warriors. If the unit doesn't have at least a decent chance of destroying the units your enemies are using, they are worthless except as military police.
Only build what you can actually use. You have no business building a library in a city that only produces 2 science. Don't build a marketplace in a city that only produces 1 gpt of tax income. You don't need a barracks in every single city; select one or two cities with access to lots of shields, and have those build all your military. You don't need a granary in every single city, either--only in "setter/worker pump" cities. I've never had a use for a granary in every city; even without them, my cities always grow right up to the riot control limit pretty quick. Usually I only build 2-4 granaries per game.
#5: Try to avoid using the happy slider, unless you're comfy with your empire and are willing to spend the extra gold to build up your score. The happy slider is surprisingly inefficient at making people happy! Luxury goods are far better. Even paying another empire gpt in exchange for luxuries is more efficient than the happy slider! (In the late game, I've discovered that turning up the happy slider ONE notch almost always costs twice as much as buying luxuries at 100 gpt from another civ! The only downside is that a potential enemy gets to use the gold you pay them for the luxuries)
If you have ONE city that's threatening to riot on you, don't use the happy slider. Turn one citizen into an entertainer. If you REALLY want to use the happy slider to prevent riots, check the loss of tax revenue (i.e. see how much it will "cost" you to turn the slider up) then see how much it costs to use entertainers instead.
for a bit, then once you get that outta your system 
build the worker first.
I am in the 2nd age at 560 AD and ive met all civs. I can crush anyone with my economic might. My military is lacking in offense but the defense is just insane 
i am in monarchy and it works really well for me. my best city is producing like 10 food and 13 shields 
(i think or maybe thats two of my cities(one is food and other is shields )) heres the game
