Othniel, the tips you are giving me are pure gold, I'm sure. But as a newbie that has only advanced up to the middleages, I don't quite understand you. Maybe when I read this a few months from now, it'll make sense.
I'm not exactly sure what part of my post above you didn't understand, but I'll try to clarify anyway.
The gorgeous thing about artillery is that when you use it, you are not at risk of losing a unit at that moment. You're just sitting back and chucking rocks, moldy cows, or whatever else they used to throw. You are damaging the enemy from a distance.
When you are damaging an enemy from a distance, you are weakening their ability to hurt your hand-to-hand combat troops when you finally send in the boys for the kill shot. Follow me so far?
On a high difficulty level in Civ, your opponents will likely be far beyond you in techs. That means, for instance, they might have Riflemen (defense value of 5) fortified in their cities. If you are far behind in techs, you might be fighting them with knights or Medieval Infantry (attack value of 4).
Now, in Civ, the defenders get some pretty significant combat modifiers or boosts. For instance, a defender who is fortified receives a +25% bonus to their defense rating when they are attacked. If you do not know much about combat modifiers in Civ, I encourage you to read up on them, either in the manual or in the threads here at CFC. (Sorry, I don't have a link to one at the moment).
What this all adds up to, is that in the example I gave--opponent has fortified Rifle, you have a knight--you WILL lose the large majority of the battles if you attack directly. The odds are stacked heavily against you. And when you are losing tons of units, you are being weakened and your attacks are not "cost-effective". In other words, you are getting slaughtered for not much gain. Not a good plan for world domination.
So, to even up your odds, roll up a large stack of artillery-type units and bomb the heck out of that Rifleman and the other six Rifle units the AI has in that city (the AI loves defensive units

). If you can, using artillery from a distance, "red-line" all the defenders i.e. reduce all the defenders to just one hit point apiece, you have just greatly increased your odds of winning. More importantly, you probably won't lose nearly as many units in doing so.
Once all the defenders are red-lined, or nearly so, send in the hand-to-hand combat troops, and take the city. Rinse and repeat across the map. The AI is very, very vulnerable to this tactic because they make a classic tactical mistake: they do not concentrate their forces onto a single target. They fight piecemeal instead, which is a great way to die. As the human, you can concentrate your forces, and that is a huge, game-breaking advantage.
Hope that helps.

Let me know if I need to clarify further, or if I totally failed to answer your question at all.
