A belief that the world was decaying towards Armageddon was pervasive in the later (post-1100) Middle Ages overall.

The devastation of pestilence, at least its initial couple of waves, certainly provoked some profound religious and theological responses. There are two approaches: one, that the slaughter forecasts the imminent arrival of the Antichrist. Second, that this is God's judgment on humanity's sins (pretty much what you'd expect--indecent clothing, bawdy and violent tournaments, disobedient children), and if people repent quickly and viscerally enough, they can forestall the end of the world for now.

Most famous in this latter category, of course, are the flagellants: people who practiced painful physical asceticism, in public but likely more often in private. This demonstrated their renunciation of sin on behalf of wider Christendom--they offered themselves up as suffering helpers in imitation of Christ in his Passion, hoping to save people's lives with the sacrifice of their flesh the way Christ saved humanity's souls on the cross.

Less spectacular, so less well known today, are the quieter devotions. People patronized increased Masses at Church, donated and lit candles to saints, even composed entire new liturgies (sequences of hymns and readings for Mass) devoted to repenting sin and pleading with God to lift the pestilence.

But while a theological or religious response was probably dominant in "popular" culture, we shouldn't ignore the proto-scientific approach that we see in university and eventually popular medical writings and even city chronicles (the 14th century, after all, is the cradle of ideas we corral away from those calamitous Middle Ages as "Renaissance"). University doctores of medicine (so, not practicing surgeons on-the-ground dealing with plague victims) sought the signs and sometimes causes of plague in the heavens, with the Middle Ages' particular blend of astrology and astronomy (and this would have had a divine component mixed in as well). They suggested that earthquakes released "bad air" that cause plague (again, linking two phenomena perceived as evidence of apocalypse).

But they also mobilized to fight plague. Learned physicians and the barber-surgeons legally required to aid pestilence victims wrote and read treatises on various possible treatments. They took this very seriously, even developing ideas about different cures working better in different climates. And of course, cities got better, in subsequent waves of plague, at implementing quarantine ordinances in various ways.

Further reading: For your purposes especially, OP, I highly recommend Rosemary Horrox (ed.), The Black Death. It's an anthology of translated primary sources from the plague years, including a substantial amount of religious material (sermons, chronicle passages, religious poetry, legal documents promoting pilgrimage-as-penitence).