We took one of our larger cat carriers just in case. But puppies rode home on my lap. OH drove. I feel it's a time to bond with puppy, after being yanked from Mum and siblings. We took collar and leash too, water and lots of clean up towels but never needed them. Jet slept all the way home. With Oban we had a big snow storm, the drive took twice as long, three hours total, and that little devil sat or stood on my lap all the way, looking out the window. Harbringer of things to come with him. With both my lap was covered by a thick old comforter to keep them cozy and soak up any accidents.
Check the link for how to transport at other times. When little I used the cat crate, they didn't know to stay in the back till they were bigger. I could strap the crate into my seat belt.
Center for Pet Safety, The Science of Pet Safety, Pet Product Testing, Crash Testing, Scientific Testing, Criteria, Pet Safety Advocate, PetSafety Some scary stuff at this link.