Make sure you are using catapults/cannons to bombard away the city cultural defense prior to attacking with your melee/gunpowder units. Suicide a cat or 2 on the city once the cultural defense is down to 0%, to soften up the defenders inside. Overwhelming force concentrated on one city is good advice. Take 8+ cats to reduce an ai capital's defense to 0 in one turn, a few more cats if you plan on suiciding some, and then take the city in the same turn.
Don't let an ai talk you into a war you are not ready for.
If you are not quite ready for a war, but you think an ai civ is getting too far ahead of you, go ahead and go to war. Take 2 or 3 cities and offer him peace. Continue building up your forces during the war, and during the peace, and declare war again once you are ready to finish him off.
Choose your enemies strategically. Certain ai civs are much more unpredictable and dangerous (Monty, Tokugawa, Caesar, Alex)...these should be first on your list to eliminate, even if they are currently friendly toward you.
And finally, don't forget to bring adequate force to garrison newly conquered cities, especially if your big stack is going to be moving on to the next city. Razing a city is better than leaving it for the ai to retake, but consider which civ might build a new city in the empty space this leaves. I'd rather take a momentary ding to my economy and occupy a city I don't really need, if it means I'm denying my next target civ some growth. This is very situational but you will get a feeling for when to raze and when to garrison the cities.