Spain, and Notre Dame in Paris were popular pilgrimage destinations for Europeans.

Enterprising individuals organized less arduous and significantly less dangerous pilgrimages for Londoners and other religious believers in the British Isles to local shrines in England, Ireland, Scotland, and Wales, including shrines at the major cathedrals of St. Albans, St. Andrews, and St. David's.

In the General Prologue of The Canterbury Tales, the narrator, who is Geoffrey Chaucer himself, tells the reader that he's spending the night at the Tabard Inn in Southwark, a fairly lawless suburb of London located on the south bank of the Thames across from the City of London, and known in medieval times as a notorious center of "entertainment."

The narrator intends to make a pilgrimage alone to Canterbury to visit the shrine to St. Thomas Beckett at Canterbury Cathedral, where it's believed that Thomas Beckett, Archbishop of Canterbury, was murdered in 1170 by four knights at the suggestion, or possibly at the order, of King Henry II.

The Tabard Inn was a popular gathering place and stopping-off place for pilgrims on their way to Canterbury from London and other parts of England.

That evening, a group of pilgrims who are also going to Canterbury arrive at the Tabard Inn.

There came at nightfall to that hostelry
Some nine and twenty in a company
Of sundry persons who had chanced to fall
In fellowship, and pilgrims were they all
That toward Canterbury town would ride. (General Prologue, 23–27)

The narrator is apparently asked or allowed to join the group, which plans to set out for Canterbury early the next morning.

So had I spoken with them, every one,
That I was of their fellowship anon,
And made agreement that we’d early rise
To take our way, as to you I’ll devise. (General Prologue, 31–34)

"The Beckett Way," a modern reconstruction of the pilgrims' journey to Canterbury from Southwark—where only the inn yard of the Tabard Inn still remains—can be found at https://britishpilgrimage.org/portfolio/pilgrims-way-to-canterbury/

Religious pilgrimages were very common in pre-Reformation Europe. The faithful would gather from miles around to converge on shrines, where they would pay tribute to a saint or other important religious figure by offering up gifts, often quite expensive ones. It was widely believed that by doing this, pilgrims would receive less time in Purgatory, or that they would be cured of some illness or other.

It is the latter reason that has drawn Chaucer's pilgrims to Canterbury in The Canterbury Tales. As Chaucer tells us in the General Prologue:

And specially, from every shires ende
Of Engelond, to Caunterbury they wende,
The hooly blisful martir for to seke,
That hem hath holpen whan that they were seeke.

In other words, pilgrims travel from far and wide to the shrine of the “holy, blissful martyr”—that is to say, St. Thomas Becket—to pray for healing and offer praises to him for helping them through illness.

St. Thomas Becket was a very popular saint in medieval Europe. He was once the Archbishop of Canterbury, the most senior churchman in England. After he fell out with King Henry II, a group of Henry's knights went to Canterbury Cathedral, where they brutally murdered Becket.

Fifty years after Becket's death, Canterbury became the location of a popular shrine. Pilgrims, such as the ones in Chaucer's tale, would come from miles around in the hope that they would benefit from the saint's alleged healing powers.

This answer can be found in the first 18 lines of the prologue. The poem's narrator tells readers that as spring comes along, people enjoy traveling and going on pilgrimages. The narrator then gives a specific example. He says that people from England enjoy going to Canterbury to visit Saint Thomas Becket. They are visiting to see the the relics at Canterbury Cathedral.

Then do folk long to go on pilgrimage,
And palmers to go seeking out strange strands,
To distant shrines well known in sundry lands.
And specially from every shire's end
Of England they to Canterbury wend,
The holy blessed martyr there to seek
Who helped them when they lay so ill and weak

This is why the various travelers are all gathered together at the Tabard Inn. It is suggested that since they are all traveling to the same location, they should all travel together. To make the journey more pleasurable, a story telling contest of sorts is suggested by the Host. Each traveler will be given the opportunity to tell multiple stories. The pilgrim that tells the best story will win a dinner at the tavern upon return.

Geoffrey Chaucer, in his General Prologue to the Canterbury Tales, introduces an array of characters who have stopped at an inn on their way to Canterbury. The purpose of their trip is religious; they are going to pay homage to the the blessed martyr, Thomas a Beckett.  

And specially from every shire's end
Of England they to Canterbury wend,
The holy blessed martyr there to seek
Who helped them when they lay so ill and weak.

Though each of the pilgrims has a different degree of devotion to the martyr, they are all going to express their gratitude for his help in their lives.

Ironically, the innkeeper has turned the table on this holy pilgrimage, and because of his proposed wager they each have a chance to win something. Now the journey is about that rather than about the holy martyr. 

The answer to your question is given in the very first line of the General Prologue to Chaucer's Canterbury Tales. It is in April that folk longen to goon on pilgrimages. Chaucer's Middle English is a little hard to understand, but it is a pleasure to read after you get used to it. Here is a sample from the General Prologue. The word "soote," meaning "sweet," would be pronounced with two syllables and sound like "so-tah." The word woold rhyme with "roote" at the end of the second line and would sound like "row-tah " People who long to go on trips in April are attracted to Canterbury Cathedral because of the beautiful countryside and the grandeur of their destination. The most beautiful lines in this passage would sound more or less like:

And small-ah fowl-less mockenmelodie

That slepenoll the nicht with open  ee.

Here are the opening lines of The Canterbury Tales:

Whan that aprill with his shoures soote
The droghte of march hath perced to the roote,
And bathed every veyne in swich licour
Of which vertu engendred is the flour;
Whan zephirus eek with his sweete breeth
Inspired hath in every holt and heeth
Tendre croppes, and the yonge sonne
Hath in the ram his halve cours yronne,
And smale foweles maken melodye,
That slepen al the nyght with open ye
(so priketh hem nature in hir corages);
Thanne longen folk to goon on pilgrimages,
And palmeres for to seken straunge strondes,
To ferne halwes, kowthe in sondry londes;
And specially from every shires ende
Of engelond to caunterbury they wende,
The hooly blisful martir for to seke,
That hem hath holpen whan that they were seeke.

Anyway, the time is April, the beginning of spring. The long, cold winter is over and people want to get outdoors to enjoy the beauty of nature.