Dante's Commedia is a narrative of pilgrimage. As with a real pilgrimage (often taken to shrines, altars, and churches), Dante walks from place to place (and flies, in the case of the Paradiso), gaining spiritual knowledge, receiving guidance from spiritual figures, and seeing spiritual sights. His pilgrimage is both literal and allegorical: in the narrative, it is an actual journey; on the level of interpretation, it is a journey through abstract concepts, through different sins and virtues. As such, it may be no surprise when, near the end of the Paradiso, Dante writes that "as a pilgrim, in the temple of his vow,/content within himself, looks lovingly about/and expects to tell his tale when he gets home,//so, through the living light I let my eyes/range freely through the ranks, now up, now down,/now circling freely all around again." Note that, in this epic simile, the vision of the highest sphere of heaven becomes the "temple" to which Dante has taken a pilgrimage; the climax of the Commedia and the final stop in the poem are thus the completion of Dante's pilgrimage. But this is a "temple" that Dante is sure to "tell his tale" about; in other words, telling a story of one's pilgrimage seems integrated into the medieval concept of pilgrimage. It's assumed that Dante will tell those on Earth what he has seen.
But we should note that Dante's Commedia was not the first time that he used pilgrimage as a motif. In his Vita Nuova, in fact, Dante explores the figure of the pilgrim in ways that might expand our understanding of pilgrimage in the Commedia itself. In one poem in Vita Nuova, Dante address "you who travel on the road of Love." Soon after, he writes: "As I rode out one day not long ago/by narrow roads, and heavy with the thought/of what compelled my going, I met Love/in pilgrim's rags coming the other way." Love, then, had a long-lasting association with pilgrimage; Love becomes both the road—making it a journey, a process—and the person taking that journey. Perhaps then we can think of Dante the pilgrim, in the Commedia, as taking the "road of Love" through Christian spirituality. Dante himself might even be seen as "Love" himself, a pilgrim in rags at the beginning of the poem, but draped in Divine Love at the Commedia's very end.
Yet, this is only one of the many possible ways of thinking about Dante's use of pilgrimage. His geographical scope, referencing areas all over Italy, his allusions to holy sites and items, and his varied language all work together to suggest that Dante himself has traveled all over Italy. One could also look at actual pilgrimages, made for specific saints, and consider whether Dante is alluding to those Saints for their associations with pilgrimage. There is simply a wealth of possibility in considering Dante's varied use of one significant aspect of medieval Christian practice.