Some general tips that prove useful for me. They may or may not work for you.
1) Try to curb your exploration. Do you really need that Galley, when you can build a Galleon not to long from now? Galleys are slow and limited. Galleons are still slow, but not as much, but most importantly can sail the sea. I would say that it is only useful to build a Galley when you know for certain that there is an artifact very close by.
In the same vein, try not to explore too much with your Warrior(s). Yes, grabbing goodie huts and defeating barbarians is good, but the adage 'knowledge is power' is not exactly suitable for the early game. The longer you go without meeting your neighbors, the longer you go without them declaring war upon you.
2) Expand, but cautiously. Build settlers. Found new cities. Take the best spots near resources. Just don't expand too far away, as it makes it hard to protect your cities. Roads obviously fix that, but until you have large amounts of disposable income, that money is often better spent rushing buildings and units.
3) Be smart about building your units. When founding a new city, I like to rush a single archer, build a second one normally, then assess the situation. If the city is in no danger, I start work on a barracks. After that is built, I build the third archer and combine them into a Veteran Army. If the city is in immediate danger, then I'll proceed directly to the third archer for the added protection, or bring a vet archer up from a different city.
Not all buildings are worthwhile to build right away. Do you need both a market and a library in the same city? No. Do all cities need barracks? No, since only one unit of the three forming an army needs the Vet bonus for the army to gain it. Are you really going to build that Pikemen unit when Rifles will be unlocked in three turns? You probably shouldn't.
In the similar vein, Wonders are often hit and miss. You might think the Great Wall is super for delaying the AI, but by the time you build it, it'll likely be obsoleted by the fast AI teching. Pyramids are hit and miss; Fundamentalism is good, but significantly slows your tech rate. For the most part, concentrate on getting the core buildings out like Libraries and Barracks, then push out units.
4) The best defense is a strong offense. Most of the time. Early game, archers are good for defending cities. However, point for point, Legion armies are equal to them in offensive measures. Given enough time, the AI will probably over run your cities; or worse yet, stall you into an endless cycle of producing new troops in cities just to hold them.
Take advantage of your offensive troops. I really like Horsemen for this; they come early and you can often get a few free from Barbarian huts. They're designed to destroy Legion armies. Double the movement means they'll always get the attack, and Legion are horrible at defending.
Horsemen are also great for stealing Settlers. Park two on a hill near an enemy capital and wait. Eventually he'll send a settler out, usually with a very meager defensive unit with it. Destroy the defending unit with the first Horseman unit, capture the settler with the second. Nothing slows the AI down faster than this. In one game I managed to do this to three different AIs within five turns, and they were never able to catch up to me.
5) Use your Great People wisely. It isn't always smart to burn your Great People right away. Sometimes, yes, but not always. A Great Builder is often best used waiting for a strong Wonder to build, such as the Samurai Castle or Hanging Gardens. Maybe settling that Great Scientist is a good idea, but first check the Tech Tree. Perhaps Irrigation hasn't been researched by anyone, and by burning him for it you get the +1 Population to all cities bonus. Which would serve you better?
That's probably my issue, I'm too dominance focused...
