Here's my own experience but there's probably better advice in the strategy section.
First kill your enemy's spare units (the one's not defending cities) which he'll be instantly sending in your direction when the war begins. If he's been isolated on an island or at peace for a long time, there will be a lot of them on demigod. I think that the AI usually wants to sends these units, and newly built ones, towards the closest enemy by land (now and then he'll send ships too but only with two-three units) so you can avoid lots of fighting by
allying with his close neighbours (closer to his capital and main cities than you). But he also considers terrain, probabilities of capturing and winning etc: if you
put a spearman on a choke-point mountain he'll probably take another route (or come by sea if there's no other route, which is much easier to defend against if you don't want the war), or if you
lure him with defenseless citys or units he'll surely send all his troops that way. Use it to get his troops to where you want them (where your troops are) and kill them with catapults and your best attackers. Use a move-2 unit to kill the last one in an enemy stack and retreat after, or you'll have to leave the last one alive. Do it in your own territory with the advantage of roads and being able to rest. In his territory HE can move 3 steps per turn by road but you can't, and HE can rest but you can't.
With the spare troops dead you can expect two defenders (enkidu warriors/spearmen/impis/hoplites/numidian mercenaries/pikemen/swiss mercenaries...) per normal city, that's what the AI uses, but more in the capital. Sometimes a unit or two may be resting there too if you let them get away earlier, but they shouldn't be much trouble. Sometimes there's a unit just built or passing by.
- Bring enough catapults to take the defenders of any city (that you want) down to 1 HP.
- Bring the strongest attackers available. Even when fighting 1 HP units some of them will be injured; have a few extra so you don't have to wait for them.
- Defenders to protect your catapults. If you lose them you're screwed, they're worth a lot of shields. The defenders also protect the attackers so that they're at full HP when you arrive at a city. Have a few extra defenders too so you don't have to wait for the wounded ones.
- Some extra of every unit type to deal with the units that the AI builds and sends at you while you're "expanding". The more enemy cities and the harder difficulty level, the more will be built each turn.
- 3 normal workers or 1.5 industrious to build connecting roads when needed.
Keep producing military and send them along the roads to protect the captured cities or to replace casualties. If you defend your catapults you shouldn't lose any, if you have enough catapults you should lose very few attackers and if you
make your first military leader into a defender army (3 spearmen or better) and walk through hills and mountains your stack will probably not even get attacked.
Catapults have an attack power of 4. In the best case you'll face two fortified regular units with def-2 and no terrain bonus = def-2.5, which should give each
bombing a 61.5% success rate, and you'll need 4 successful bombings to get them both down to 1 HP. The number of catapluts to bring depends on how much you want to gamble. If your catapults have bad luck you may have to consider waiting a turn and try again instead of risking your attackers lifes.
In the worst case you'll face two fortified veteran units with def-4 on a hill with walls, which gives each bombing a 33.3% chance of success (about half of the best case, so bring double the amount of catapults) and you'll need 6 successful bombings to get them down to 1 HP (1.5 the number of catapults).
Don't kill only one defender, the AI will rush a new one. Kill them both in one turn.