Total War: THREE KINGDOMS

This is listed under The Furious Wild DLC, but it should be completable without. This achievement has since been moved to the title update section.

This achievement might be glitchy in our favor? I got this almost immediately in my first campaign, but I'm about 85% sure there were no elephants in the other army. The other possibility is that there's a Han general that starts with an elephant mount, but I've never seen him across all of my games if that's the case.

In case it's not, as far as I'm aware, only Nanman have access to elephant units. The Nanman are all clustered in the southwest, so Sun Jian and Shi Xie (free DLC) are probably the closest selectable characters. You may want to scout their armies first and make sure they have an elephant unit deployed before declaring war on them.

For fighting them, I found Nanman to be kind of annoying and the elephants more so, even on the easiest difficulty. It shouldn't be too bad if you set the difficulty down, but I'd still look up some anti-Nanman strategies before going for this achievement (Besides, you still need to win the fight, not just kill the elephants). From what I remember, elephants supposed to be weak to ranged attacks, but personally I've found them to still tank quite a few arrows. Still, it's the most reliable way of dealing with them that I found, so bring plenty of archers and focus fire. Nanman are weaker to fire as well, so fire arrows and flame shots are a good idea. I think the infantry can be dealt with similarly to Han infantry, and I've heard tigers are susceptible to counter-charges. Nanman get quite a few bonuses when fighting on their territory, so it may be easier if you can lure them out to you instead.

Han generals can equip elephant mounts, but the only way I've seen of getting them is through a merchant event where you can buy one for 2000 or 4000 gold (I don't remember which one exactly, just that it's a bit pricey), and I have no idea if AI can utilize the event. You could try buying an elephant and trading it to another faction that you'll declare war on later. It's a lot more RNG reliant, but it might be easier to fight one within a Han army than a whole unit or more within a Nanman army.