Read through a couple of the sucession games, particularly training games (e.g.
http://forums.civfanatics.com/showthread.php?t=221622) those should help.
Grow population fast
The difference between a city with 3 extra food and 2 extra food is huge, so if your worker can get you that extra food he should. Improve cows and wheat first, and work that square, then any other square that can get you 3 food or more. This means that workers matter, and you should concentrate on what you are doing with them.
Use the Lux slider
You should try not to use entertainers in cities. Instead use the Luxuries slider on the F1 screen, set it to the right level to make sure everyone can work without rioting (happy citizens must be equal to or greater than unhappy ones). As you get to multiple cities that may get more difficult, so you might need to use the odd warrior in a city, or if a warrior can't get there in time the odd scientist.
Don't be afraid to let this get to 40% or 50% in the early game, but as the game progresses it should settle down to 10% to 20% if you have acquired sufficient luxuries. Don't forget to check this each turn, you don't want to waste the gold, or have cities riot.
Build more workers
One per city or more is what you are aiming for, until you have no cities working tiles that are not fully improved.
Dont defend every city
You don't need a military unit in every city, you will need some to protect from barbarians, but it might be easier just to spend your gold, and let them waste themselves on your cities.
Don't build every building in every city
You only need baracks in cities that will produce military, granaries in cities that are producing workers or settlers, and probably not much more infrastructure until you get out of the ancient age apart from a couple of libraries in high commerce cities
Trade
Technologies are more valuable the fewer people that know them. So don't buy a tech that only one person knows unless you really need it. Wait for your opponents to trade it round first.
Similarly if you have a tech that one of the other AI's has trade it round to all the others before they do, you might as well get something for it rather than them.
Get out of despotism as soon as possible.
The despotism penalty (if a square is producing 3 or more of something it loses one) is huge.