Follow the airline's guidelines on marking the carrier, feeding and watering instructions, etc. At your destination, wait until after you leave the airport before allowing your animal out of the carrier. Small and medium dogs are safest in a carrier as well, and the carrier should be secured with a seat belt. For large dogs, some vehicles can be outfitted with a metal grid which keeps them in the back. And you can actually find special seat belt harnesses for larger dogs.
Some pets get carsick, and vets can prescribe travel-sickness medicine.
Don't leave your animal alone in the car in hot climates. Temperatures can quickly exceed 100 degrees, even with some ventilation. Or very cold climates. Cars are not well insulated and temperatures can drop very quickly. If you suspect your animal has heatstroke or hypothermia, get to a vet immediately.