I have found some answers about medieval time lodging and travel that may interest you:

• Did public inns or taverns exist in 11th-14th-century Western Europe? by u/Naugrith
• I'm a traveler in ad1200 going from Lübeck across the HRE to Rome on secular business; Do I travel alone or in a group? Do we camp every night in the woods or am I in a Tavern/Inn every night? by u/PartyMoses
• I'm a peasant visiting a 14th century alehouse. How would I be charged for my drinks and how easily could I skip the bill? by u/sunagainstgold
• How Were Medieval Taverns and Inns Owned and Operated? by u/sunagainstgold
• I am a peasant in 13th century Europe, living in a reasonably large town. I want to visit my brother in the nearest large town. How long will the trip take me? What kind of dangers will I face? Are there inns or farms along the way or will I sleep outdoors? by u/LouisMarcy
• In books and movies, when a knight stays at a tavern, he leaves the next day on a “fresh horse.” How did innkeepers keep track of which horses they owned, and how did they get them back? by... guess who... u/sunagainstgold!!
• Which time period did drinking tavern/inns become a thing in Europe as depicted in fantasy stories such as LOTR? no direct answer was given but u/grumpenpole linked several answers.

Some of these posts contain information about other stuff which is also very interesting on its own, so I recommend you to press "view all comments" and read the other threads!