It would take a pilgrim about one to two weeks to walk from London to Canterbury in the 14th century, depending on their pace and rest stops. The journey was approximately 60-70 miles by foot.
It would depend on the specific pilgrim and the purpose of the journey. Each pilgrim in Chaucer's "The Canterbury Tales" has their own unique personality and quirks, so the best choice would be someone with whom you would enjoy engaging in conversation and sharing stories.
The Pardoner in Geoffrey Chaucer's "The Canterbury Tales" is the pilgrim who did a brisk business in fake holy relics. He would sell fake relics to unsuspecting people, claiming they had miraculous powers.
All of the characters in the Canterbury Tales are pilgrims, and the main reason they are traveling is to visit the shrine of Saint Thomas Becket at Canterbury Cathedral. To be more specific about individual motivations though, you would have to specify a character by more than "pilgrim."
That would be London.
Tower of London
The original plan was that each pilgrim would tell two stories on the way to Canterbury, then two more on the road home to London. But Chaucer never completed the work, and most of the Pilgrims get to tell only one story. (In the Canterbury Tales as we have it, the pilgrims never arrive in Canterbury, let alone begin the journey home).
If you mean by "chronicler" a writer of history books on the subject it would probably be Barbara Tuchman, whose book "A distant mirror" focuses on this century and was a massive bestseller back in the eighties. If you mean a chronicler who lived in that century, Geoffrey Chaucer, writer of the Canterbury Tales, would be the one.
The Canterbury Talesis about a group of pilgrims who venture onto Canterbury for a "pilgrimage"which most used for a break from their usual labors and happen to meet TheHost in the Tabbard Inn who proposes a challenge for the most entertaining& moral tale to be told in return for all the pilgrims to pay for the best mealfor the best tale. Each would tell two from and back Canterbury but mostwere never completed or are rumored.
Madame
To worship a cow.
They would hang out at the Jamestown Mall, play tennis, watch soap operas and go to the Pilgrim Beauty shop once a week, go to the Pilgrim "Stop and Shop" for their groceries. Bowling at the "Pilgrim Bowlarama" and once in a while take in a movie at the "Pilgrim Muvico"[only one screen back then]
On the pilgrimage to Canterbury, pilgrims would have had two choices. Either go by foot or horseback. They followed an unpaved path to the shrine of St. Thomas of Beckett. Due to the roughness of the environment and period of time these journeys were taken, other modes of transport were not available.