This achievement is pretty simple to get and I managed it on my first encounter playing as the Minstrel.
Simply start a game playing as the Minstrel. When one of your encounter cards is an animal, you can choose to charm it rather than fight it.
To charm an animal you simply have to beat it in battle as you would normally - but instead of losing a life/dying it joins you as a follower.
Obviously the stronger the animal the better a follower it is but with regards getting the achievement you can just take whatever you find that's beatable. I got a Boar with a strength of one for my first fight so pretty much got it automatically.