All you need for this is someone with Animal Whisperer. To "charm" them means either getting passive animals to follow you or making animals run away during combat. Having even 3-4 points in the skill is enough for the early animals you come across, and there are quite a few dogs, chickens and goats scattered around the opening villages that you can recruit to follow you. Make sure you charm the 5 goats on Goat Hill near the prison, which will also unlock the "Goat Herder" achievement.
If you don't want to spend skill points on Animal Whisperer, you can recruit Ralphy in the Rail Nomad Camp if you save him from drowning when you first enter the camp. A short time later his mother will ask if he can join you, and he comes with some points in the skill (although not many).
If you're having a hard time, you can boost the skill another point with a trinket found in a diggable spot near the exit to the Wasteland in the Ranger Citadel.