Upgrade Your Troops
The best part of earning promotions is that they stay with the unit when you upgrade them. That means as soon as you research a technology that allows you to build better units, you should throw down the gold and get better troops. Once you research Iron Working, and secure a source of iron (see above,) your lowly Warrior can become a Swordsman and wreak havoc. When choosing which technologies to research, you should make a beeline for those that help your armies first, especially if you know that you have a bunch of units that can be immediately improved. Upgrading does cost a significant amount of gold, but that's what you've beating up the barbarians and sacking cities for anyway, right?
Once you get Chivalry and can upgrade your Horsemen to Keshiks and Spearmen to Pikemen, then the rest of world will tremble from the sounds of your army's hoofbeats.
Attack in Mixed Groups
It's important to have an army that doesn't have any weaknesses. As powerful as they are, invading with only Horsemen will end in defeat and disgrace. Teaming the fast cavalry units with slower foot soldiers like Spearmen and Warriors to absorb counter-attacks is the only way to win. Spearmen are especially effective against mounted units such as enemy Horsemen so it's imperative to bring them to protect your flank. I've already mentioned how Archers and other ranged units are important, but these must also be protected.
Promotions should also match a unit's characteristics; there's no benefit in sending a Horseman who's been promoted to succeed in Open Terrain with the Shock promotion to attack a Spearmen in a Forest. That Horseman is just going to die, and not very honorably. Send in the Swordsman who has the Drill promotion (bonus in rough terrain like forests) instead.
Strong Navy Wins Wars
Even though Genghis Khan rampaged Eurasia without a navy to speak of, naval superiority in Civ V is awesome and shouldn't be overlooked by any self-respecting warlord. Not only is a navy important for protecting your transports, but they work great as very mobile artillery units. In the early game, Triremes are only effective at plundering fishing boats, depriving your enemies of vital resources, or firing for minimal damage on enemy units. But once you get Caravels and Frigates, you can lay down some serious carnage on unsuspecting enemy units from serious range. And when you get the Range promotion, allowing you to bombard from beyond the range of a city's own mortar attacks, your navy can clear the way for your invading force quite nicely.
Yes, it may not be in the character of the Mongol Empire to defeat the Greek and Aztec civilizations by using Frigates to soften up the front line as your Riflemen and Cavalry storm in, but that's what Civilization is all about -- messing with historical accuracy and crushing your enemies. And now that you have all of the tips I've mentioned above, you should have no problem doing just that.