Guildenstern
Tepegian
I'm sure something like this has been mentioned many times before; there's been plenty of time to play Civ, and plenty of players out there more skilled than I. 
Anyway, the following is the war strategy I use, which makes use of a combined force of defensive, artillery, infantry and cavalry units all:
1. I never use cavalry units (i.e., Horsemen, Knights, Cavalry) to attack cities. Cavalry units are used to protect my Stacks o' Death from the units sent at them by the Civ being attacked (because faster units have an easier time of running out from the Stack and eliminating any threats that might otherwise slow the Stack down).
2. The Stack o' Death consists of a combined force of offensive infantry units (Archers, Swordsmen, and any other unit with 1 movement and strong attack), defensive units (Spearmen, etc.), and artillery (Catapults, Trebuchet, etc.). I haven't narrowed down exactly what the ideal numbers are yet, but it seems the Stack is most effective when it has a decent number of artillery, about four or five offensive units (more if the cities you plan to attack have more units defending them than that), and at least as many defensive units as the number of cities you intend to take.
3. The strategy itself is simple: because all units in the Stack o' Death have the same movement rate, they simply march from city to city, camping outside to hurl large rocks before attacking and capturing. Stop in each city to heal before going on to the next; and be sure to leave a defensive unit behind at each city. During this whole process the cavalry units are used to fend off the enemy units that the AI sends at you, so that the Stack does not have to deal with them. If you have only a few fast units, keep them close to your Stack; if you have lots, have them run all around the enemy territory, both for scouting and to pick off enemy units that are farther off.
4. Make multiple Stacks for many-pronged wars (two have usually been sufficient for my purposes, but your playing style may be different). If your foe is a backward nation full of unroaded swamps, it might also be a good idea to throw a few Workers into your Stack so that they can build roads to get your artillery within firing range of the AI cities; in this case, put in more defensive units as well so you can protect the workers.
To be honest, I have no idea how well this works on the higher difficulties; I currently play at Regent, and it seems to do fine there. This strategy is a bit slow, as your main city-taking forces have only a 1 movement rate, but if done properly you should be able conquer while losing few if any units.
Of course, if you have the capabilities, this strategy should always be combined with resource-denial strategies (by either pillaging or by going after resource-linked cities first). Navies are useful, too, both as blockades and as scouts for coastal areas. Once you have the power of flight, however, an entire new dimension is added to warfare; this strategy is only meant for pre-flight era wars.
Anyway, the following is the war strategy I use, which makes use of a combined force of defensive, artillery, infantry and cavalry units all:
1. I never use cavalry units (i.e., Horsemen, Knights, Cavalry) to attack cities. Cavalry units are used to protect my Stacks o' Death from the units sent at them by the Civ being attacked (because faster units have an easier time of running out from the Stack and eliminating any threats that might otherwise slow the Stack down).
2. The Stack o' Death consists of a combined force of offensive infantry units (Archers, Swordsmen, and any other unit with 1 movement and strong attack), defensive units (Spearmen, etc.), and artillery (Catapults, Trebuchet, etc.). I haven't narrowed down exactly what the ideal numbers are yet, but it seems the Stack is most effective when it has a decent number of artillery, about four or five offensive units (more if the cities you plan to attack have more units defending them than that), and at least as many defensive units as the number of cities you intend to take.
3. The strategy itself is simple: because all units in the Stack o' Death have the same movement rate, they simply march from city to city, camping outside to hurl large rocks before attacking and capturing. Stop in each city to heal before going on to the next; and be sure to leave a defensive unit behind at each city. During this whole process the cavalry units are used to fend off the enemy units that the AI sends at you, so that the Stack does not have to deal with them. If you have only a few fast units, keep them close to your Stack; if you have lots, have them run all around the enemy territory, both for scouting and to pick off enemy units that are farther off.
4. Make multiple Stacks for many-pronged wars (two have usually been sufficient for my purposes, but your playing style may be different). If your foe is a backward nation full of unroaded swamps, it might also be a good idea to throw a few Workers into your Stack so that they can build roads to get your artillery within firing range of the AI cities; in this case, put in more defensive units as well so you can protect the workers.
To be honest, I have no idea how well this works on the higher difficulties; I currently play at Regent, and it seems to do fine there. This strategy is a bit slow, as your main city-taking forces have only a 1 movement rate, but if done properly you should be able conquer while losing few if any units.
Of course, if you have the capabilities, this strategy should always be combined with resource-denial strategies (by either pillaging or by going after resource-linked cities first). Navies are useful, too, both as blockades and as scouts for coastal areas. Once you have the power of flight, however, an entire new dimension is added to warfare; this strategy is only meant for pre-flight era wars.
