I've hauled them many different ways, like mentioned above.
Babies on my lap (or tucked inside my coat for the really little ones.)
Young goats in smaller crates in the back seat.
In various dog crates, sized appropriately, in the truck bed when the weather has been nice, in the horse trailer for bad weather, and even in the back seat of the truck. Not reccommended though, they don't take kindly to seat belts.
I was taking a full sized doe to be bred several years ago before I owned my own buck. The road I had to traverse needed a 4X4 to get up it and was too trecherous to even think about taking a trailer up in winter. It was also too dangerous to try the crate in the back method either, and freezing cold wind + goat = sick goat as they can't take the cold wind.
So I folded down the back seats (they fold flat like the back of a suv or mini van) spread a tarp out (for moisture control) and covered with an old blanket.
A friend drove while I held the goat in the back. Many goats will lay down while in motion, but some will try to jump up front and do the driving. That's why it is not reccommended without help!
Like horses or dogs, some love car trips, others hate them.
I've also hauled them and my alpacas (not together though) in the back of my SUV.
Boy the looks I get when traveling down the road with alpacas is hilarious.
No body can figure out what the heck kind of dog is that??!