From an earlier answer of mine:

Account records for the travels of warden and other officials from Merton College, Oxford, exist from 1270, and G. H. Martin has a great article laying out the itineraries and purchases from some of the better recorded years (mostly fourteenth and fifteenth centuries).

There was typically some stocking up of provisions from the outset. In 1325, that meant fish, candles, ale, and spices; in 1330, cheese, candles, spices, and oats (probably for the horses). The spices, we find out later, were likely gifts for the royal officials and nobles being visited. One thing to keep in mind is that this is just things purchased. Vegetables grown in college gardens and perhaps grains/bread from grains grown on their outlying farmland would not appear in the accounts, as they did not have to be bought.

Our warden traveling in 1330 did not do a great job with his calculations, it seems, because he had to stop on the very first day to buy more oats and hay for the horses. On the second day, his party stopped partway through the journey to buy drinks. That night, they ate a simple meal and loaded up on provisions for the coming days: bread, ale, fish (herring), and more hay and oats for the horses.

The third night, they had a feast! Martin notes eel and flounder both on the menu, served alongside garlic and onion (now that they are far from Oxford gardens) and even a little bit of fruit. Oh, and wine.

Travel food would of course have had a strong class component; we're looking at fairly well-off people here. Looking at Crusades chronicles, there are references to knights being reduced to eating barley (bread), the peasants' grain. This suggests that peasants would continue their pattern of eating pea flour bread or barley bread, BUT knights were used to still consuming varieties of wheat flour bread even on (less arduous) travels.

The Merton account books also give a taste of some of the unexpected purchases necessary along the way: new saddles, new horseshoes, new spurs, vet care; alms for beggars or pilgrimage sites; new clothes or blankets if it rained too hard; and in 1325, repairs to the warden's sword. I would love to know the full story there!

As for staying overnight, inns (formal or casual) dotting the roads and cities were a common host for the Oxford travelers and others. The only reference I've personally seen to people sleeping in tents is outside Rome during jubilee-esque pilgrimage years, but that doesn't mean it couldn't happen in the course of normal travel. Monasteries also often had guest houses or rooms.

The warden's party in 1330 additionally spent some nights at the manor houses of noble patrons of the university. They brought as gifts items like the spices mentioned above and others purchased along the way--almonds, sugar, ginger (the latter two generally being considered spices, but singled out in the records).