I chose Ticinus for our first scenario: there aren't so many different units to learn so we were able to get underway much faster.
Joel, my roommate, advanced some Carthaginian light cavalry on my right. I sent some Roman medium cavalry against the corner of his formation, and after much confusion, flipping through rules, discussion, and redos, we finally decided that I probably destroyed a unit.
Joel then advanced his ferociously powerful Carthaginian heavy cavalry up the middle, and my light infantry fled for their lives out of his charge range but made holes in the line for my medium cavalry to get through. He then split a wing off and counter charged the mediums I had still hanging around his light cavalry, wiping one unit out and causing the other to limp wounded back to my lines.
With his cavalry split, I charged Scipio and half of my mediums against one of his wings. The attack hardly made a dent, and his fierce counter punch killed Scipio and wiped out a unit. Fortunately, I was able to use "Evade" to save the remains of my charge, but there wasn't much left of them.
There was good news, though. His counter charge left his formation in disarray, and I had yet another "move all mediums" card. This time I was able to completely overwhelm heavies isolated by the fighting. He responded with "I am Spartacus" but had spectacularly bad luck on his roll, and was limited to moving only one unit in response. I then played a THIRD "move all mediums" card, and this was enough to destroy all but one of his heavies. I paid for it, though, as his ability to "battle back" reduced all but one of my mediums to single strength.
A cloud of his light cavalry then galloped out and peppered my tattered mediums. We fell back under a hail of arrows, but the arrows were too flimsy to inflict any serious harm. One of his units rode too far out trying to encircle my mediums, so an unruly gang of my light infantry mobbed and destroyed it. These maniacs then went running after his other light horsemen who scattered like the wind before them. The poor fools waddled their way across the entire map, trying to to trap one of the elusive light cavalry, but they walked into a determined counter charge that put an end to their shenanigans. Once again, though, the flimsy weapons of the light cavalry didn't do anything decisive.
My light infantry was now running for their lives with as much energy than they charged with, his horsemen in pursuit. What he didn't see was that my medium cavalry that had sneakily reformed, and he walked into their counter charge which was enough to secure the victory for me.
I was assisted greatly by my cards, but even so, we concluded that he could have still won if he had supported his heavies with his lights rather than committing them separately.