I am surprised there is not a "combined arms" article in the "War Academy". This is a term that I have seen often on this site, but seldom fully explained. The "combined arms" use of Infantry, Artillery, and Cavalry are classic. Though you can't replicate the real-world use of combined arms in the game ("screening" and reconnaisance use of Cavalry in the game are ineffective and wasteful) you CAN use the principle of combined arms effectively.
There are a couple of keys to warfighting in this era: First, technology favors the defender - Infantry are formidable in the defense. Second, movement generally favors the defender early in the war if he has a rail network built up - though this will work against him after you capture territory and can reinforce your position on the captured rail network.
Many are quick to note that Infantry and Cavalry are numerically equal in the attack - but the retreating ability of Cavalry makes them MUCH more powerful - particularly if you're fighting overseas. Each unit that survives to the next turn is one less unit occupying precious transport space - even if it has to rest a few turns in cities without barracks.
A common complaint about the tactic of bombing enemy cities with artillery is movement. While moving one square per turn in enemy held territory is slow, there are some ways to offset the downside of sluggish advances.
1) Take the high road. If you can take a path to the next enemy city that keeps you in the mountains, TAKE IT! Your infantry will enjoy a nice defensive bonus, and your cavalry won't be tempted to outrun your infantry. Your Combined Arms Task Force will arrive at the city's gates at full strength, ready to do some serious damage.
2) Consider opening up a second (or third) offensive axis of advance on the same front. If you have the resources (units and movement ability) to assemble additional CATFs of Art/Inf/Cav, and start a second axis of advance. If you have 3 CATFs operating in concert, one can be advancing on a city while a second is attacking a city, and a third is consolidating a city just taken.
A lot has been said here about the need to fight short wars when you are in Democracy or Republic, and these tactics do not lend themselves to short wars. It is important here to distinguish between a LONG war and a PROTRACTED war. If you are shelling a city with artillery and attacking with infantry and taking 3-4 turns to take a single city, you run the risk of a PROTRACTED war of attrition. If, OTOH, you are taking cities in 1-2 turns on each axis of advance, and KEEPING those cities after taking them, you will be fighting a much more decisive war. If part of your overseas army is consolidating in conquered territory, this should not contribute to war weariness in your homeland.
By maintaining the initiative, and giving the AI no rest... usually the war will reach a turning point where the enemy's army is sufficiently weakened that it makes more sense to send waves of cavalry into the remaining cities to finish the AI off in more of blitzing fashion. At this point the cavalry losses are of less consequence than ending the war - but the key is to END the war this way.
Some of my personal notes on this strategy:
Unit mix for a CATF - 2 or 3 parts cav to 1-1.5 parts Inf to 1-2 parts Arty. A transport load is the basic "part" for my recipie.
On logistics - I usually deploy my artillery in stacks with an equal (or greater) number of infantry. Sometimes I need to sacrifice an infantry or two to take out that last defender, and of course, you have to leave at least one behind after you take it...sometimes I'll also string unbroken lines of infantry across portions of the front, to keep his Cav off the rail networks outside my cultural boundaries (if left uncovered he could hit me deep, where my wounded units are resting).
On wartime production - Once I have my third CATF, I quit sending Artillery into the theater. (I do keep a stack at home, just in case). During the war I'm making CAV to replace losses, and infantry to replace the ones I've had to leave defending captured cities. It's just a pain to augment a stack in enemy territory, and I don't want to waste a turn keeping a stack still to reinforce it.
On pillaging and shelling roads - I don't do it. It hurts your diplomatic relations with the other AI civs, and is a distraction from the real military effort. My tactical goals are to destroy the enemy's army, and make his cities mine.