Need some more details like your attack dates, and the kind of the AIs you are targeting (a Protective AI is MUCH harder to hit with early units and can shut down everything until siege units sometimes, certain AIs build a lot more units than others etc.) just to get a general picture of your timing and standing. On Noble you should be quite able to push the nearby AIs over just by expanding quickly enough and running them over with masses of your unit of choice if you hit early enough, there's not a lot of economic punishment on this setting so you can just keep cranking from 3-4, as much as 6 cities indefinitely if you have the happy to support it.
A good general tip though is to learn the diplomacy mini-game. It's hugely useful and was toned down a lot in 5/6 as I understand it. In those situations you describe where the AIs are instantly responding with stacks, you can make sure they are far away from home before you stab him in the back by getting him into a war with someone else beforehand, for example. You can bribe in a buddy and have his stack clear out some of the defense of a city before you clean it up and take it, etc. It's also pretty vital when attacking technologically advanced targets to isolate them so THEY don't do the same thing to you when you declare war.
Also, use slavery to whip your armies faster if not already. Slow building can work with very good spots but is generally inferior in term of producing lots of units over time. Chop too. Production now is much better than production later when attacking.
Even if the AI doesn't have a huge stack waiting, it will concentrate defenders in "threatened" cities to make your job harder. The AI was designed with quickly aggressive human players in mind, you see, so it focuses on defending. The good news is you can use this against it; feign an attack on one city by putting a small group next to it until you see units popping up like crazy (meaning the AI is routing them there) then use your primary force to strike at another city, or to pick off defenders that try to scramble between the cities where they are extremely vulnerable.
You don't always have to go for the immediate kill, and can instead quickly get a few nuisance units in strong defensive spots near cities to "choke" the AI. They'll freak out and build defenders mostly and skimp on developing or expanding, letting you gain a significant advantage over time where you can eventually crush them. It works very well on Noble to get an Archer or Axe on a Forested Hill near enemy cities and they'll overreact trying to get rid of it or protect against the "threat" it creates, so much so that a choke that costs you like 60-70 hammers can ruin them for the rest of the game until you decide to put them out of their misery. One thing that a Deity player I watched play reiterated many times was that if you just hang around in the AIs face long enough they make mistakes you can capitalize on, and it does work.