A really interesting question. There are two preliminary questions here, I think. First, and not entirely germane, what were things carried in? Second, who carried what?

Regarding the first, sewn-in pockets appear to be an innovation of the 17th century. Before then, by around the 1200s there were pocket slits made in the sides of outer garments that allowed access to a pouch or purse hung inside on a belt (or girdle) around the waist. Alternately, the purse simply hung on a belt worn outside the garment, as in the little purse called an almoner hanging low at the side of an effigy of Queen Berengaria, Richard I’s wife. Or it could be worn on the belt in front, like a Scottish sporran (needed for pocketless kilts) or a modern fanny pack. (Apologies if I made Brit Redditors giggle just now.) See this early 14th-century drawing of two men wearing their purses in front from the Luttrell Psalter (BL Add MS 42130, 198r) for an example. There were no pickpockets in the modern sense because there were no pockets to pick. Instead, there were “cut-purses” who cut the strings that held the purse or pouch to the belt. A 16th-century play called Hycke-Scorner has the victim of a cut-purse say “From my girdle [i.e., belt] he plucked my pouch; / By your leave, he left me never a penny; / Lo, nought have I but a buckle.”

The belt itself could hold other “things to carry,” especially daggers, carried by men and women alike. No well-turned-out man seemed dressed without a dagger at his side (note the one on the belt of the man on the left in the above illustration from the Luttrell Psalter). In his “Canterbury Tales,” Chaucer says the Franklin had “a dagger (anlaas) and a silk purse (gipser) hanging from his belt. Women as well as men could carry daggers, as is sometimes depicted on brass funeral effigies. An old Welsh ballad called "Glasgarion" from at least the time of Chaucer, who knew it, has a maiden pull “forth a little pen-knife that hang by her knee,” which she uses to kill herself after being raped.

Purses and pouches were known by several names: gipsers (gipcieres), almoners (aulmonieres), haumudeys (a corruption of aulmonier), pokes, etc. And they could be made of materials as humble as cloth or as sturdy as leather, especially boiled leather, or “cuir bouilli,” which stiffened the leather and made it suitable for stamping with designs. A 13th-century French poem has a merchant say: “I have store of stamped purses, red and green, white and black, that I sell readily at fairs.” The carpenter’s wife in Chaucer’s “Miller’s Tale” has “a purse of leather tasseled with silk and decorated with metal pearls (“purled with latoun”) hanging from her belt.

There were, of course, other ways to carry things such as in baskets or sacks, both good for taking barter to market or toting purchases back home. A specialized version of the carry-all was what was known as a pilgrim’s “scrip,” basically a satchel. Along with a pilgrim’s staff, these were what distinguished you as an “official” pilgrim. These items were often blessed in church, as in this illustration of a bishop blessing them. The scrip would contain basic items needed on a pilgrimage: food, utensils, spare clothing, money, travel documents, probably some charms for safe travel, and, especially for pilgrims seeking healing or already healed by invoking a saint, an “ex voto,” a small effigy of a cured arm, leg, eye, etc., made of wax, tin, lead, silver, or even gold to be left as a thanks offering at a shrine. Here’s a 15th-century pilgrim with his blind son at a shrine seeking a cure. Note the scrip, which the father has hanging around his neck. Note also that both the father and his son wear hats with pilgrim badges on them. These were collected from other famous shrines, holy souvenirs of a sort, which here suggest they have been to other saints’ shrines unsuccessfully.

So, finally—and what’s above is probably TLDR—what did medieval people carry? Well, obviously money—coins—if they were on their way to market for food. From the Luttrell Psalter again, here is a well-to-do man opening his purse filled with coins to give alms to a handicapped boy. (Alms, of course, is why one word for a purse was “almoner.”) I’ve already mentioned the daggers or knives that men and women alike could carry. But social status would also determine what got carried. For instance, chapter 55 of the late 5th-century Benedictine Rule specified that monks were to eschew private property, but were allowed to own: “cowl, tunic, stockings, shoes, belt, knife, pen, needle, towel, writing tablet.” The knife was an all-purpose utensil, needed for eating (no forks yet!) and for doing things like sharpening the nib of a stylus—thus, “pen knife.” That monks were expected to wear their knife regularly is suggested by chapter 22 of the Rule, where Benedict tells his monks to take the knife off before bed, “lest they accidentally cut themselves in their sleep.” Here’s an illustration from 1470-80 (from BL Royal 14 E IV f. 10) showing a monk wearing a purse on his belt, a small case for a stylus, and an inkwell, the tools that a writer would need. Benedict’s Rule expected monks to be readers and writers (though this latter skill was later for monks trained to write skillfully).

People of the lower classes would carry the things they needed for daily work: knives, of course, but also gloves tucked into belts, hand scythes, flails, buckets, and other farm implements. People of the upper classes and, by the later 14th century onward, middle classes, especially women, might also carry small prayer books—called Books of Hours—attached to their belts. These were called "girdle books," as seen at the waist of the young Elizabeth Tudor. Women who headed a household would carry keys. For instance, images of St. Zita of Lucca, patroness of maids and servants, often show her with a ring of keys on her belt. Another popular devotional item that middle and upper class people of the later Middle Ages might wear is a set of rosary beads, called “paternosters.” By the 14th century, people with poor eyesight might carry around newly-invented spectacles. (Over 1,000 pairs were imported to England during a 3-month stretch of 1384).

Sources: I’m relying on manuscript images to a great degree here but trying to keep in mind Michael Camille’s caution about taking what we see in manuscript illustrations, especially the Luttrell Psalter, as accurate representations of reality. See his Mirror in Parchment: The Luttrell Psalter and the Making of Medieval England (1998). For purses, pouches, and pockets, etc. I’m drawing on older studies of the evolution of medieval English fashion, such as Frederick W. Fairholt’s Costume in England from 1896 but still a useful compendium, and Herbert Norris' Medieval Costume and Fashion (1927), but also Rebecca Unsworth’s “Hands Deep in History: Pockets in Men and Women’s Dress in Western Europe, c. 1480–1630” (2017).