Bighorn Sheep, because it sounded like an interesting animal to be.
The changes are immediate. Your hands harden and fuse into hooves while white fur sprouts all over your body. You feel a pressure on your skull and ram horns sprout. Your ears reshape into goat ears. Your skull elongates into a goat head and your eyes move to the side of your head. Your pupils become horizontal lines as your tailbone extends into a tail. Your bones shift until finally you're on four legs.
You shake out your new limbs and raise your eyebrows. They're very flexible. You step out of your stall to find that you're first. A few minutes later, your wife steps out of her stall and your instincts tell you to RUN! She's a Snow Leopard, and Snow Leopards eat Bighorn Sheep. You hold back your terror as she smiles at you. You smile back. Next to come out is James, who's a Golden Eagle, and finally Lilah, who's now a Llama.
"This is so cool!" James says, spiraling into the air. You smile. "Good to hear." You say. "Now, let's go into the mountains." Your wife says. "Okay!" Lilah and James say in unison.
You open the back door that leads to the enclosure by stepping on a pressure plate. Your whole family runs (And in James' case, flies) into the enclosure.