well on noble at least, i think your first priority should be to take the enemy capital first with a quick ancient style blitzkrieg. The reason? its most likely his strongerst military production city, where he is most likely to whip as many archers as fast as possible.
can't say if this is true for prince, but I remember playing a bit more warlike bismarck quite succesfully. First I archer rushed mansa musa, 8 archers I think against 2 warriors and a skirmisher

. This won't work on noble+ levels.
but after that I decided to go bronze working and iron working if by chance I don't get any metals. I eventually got both resources, and I moved a mighty stack of 6 swords and 6 axes as close to his capital as possible, and I indeed took it, although all my swords died in the process. A couple of bowmen defending there and so on.
So when rushing, BEWARE of enemy's unique units that come early, such as dog soldiers(american indians), bowmen(babylon), skirmishers (mali). Also preferably target weaker civs, that is those civs that are not protective or aggressive.
Although I must say that dealing with montezuma and shaka earlier is better than later.
So in a nutshell pick your target, build an army, move for the killing blow (take his capital) and only then worry about the rest of his puny resently settled cities.
Ive personally found out that only if you wait too long around the capital trying to pillage stuff and not assaulting right away, then the AI starts to whip a ridiculous number of archers in his capital, not good. Generally they manage to whip around 2 archers or something like that while you walk your stack into attack position.
and pray to god they dont have axes defending also

. rush only if they dont got metals