First off peasants did not travel. It was illegal since they belonged to the land they were born on. They had to have their lord's permission to leave their home village/farms. When they did get it, they walked.
Craftsmen and tradesmen and peddlers used rough carts or packed things on their backs. A few might have had pack mules or OLD horses close to the end of their lives.
Churchmen (clergy), officials, and upper class merchants and musicians would have palfreys (light boned horses) or mules to ride. Most of the clergy, though, walked. So did musicians.
The nobility had the best animals. They ranged from heavy-boned warhorses to palfreys to hunters. When they traveled, their goods were in wagons and they rode horses if at all possible.
The carts and wagons, even the best of them, had no shocks or springs in this era. The most comfortable way to travel was by horse litter (slow, but more comfortable than a wagon) or boat. Whenever possible, people took to the water to travel even though most could not swim and boats were dangerous.