Rosemary Horrox, author of the very useful source book The Black Death (1994), establishes three categories of then-responses and explanations of the Black Death. Religious (theological explanations), scientific/natural (such as astrological, 'seismological' or miasmatic explanations) and human agency (the idea that someone engineered the plague). My interest and knowledge primarily lies within the last category, the notion of human agency as the cause of the plague. But pursuant to the rules of digression of the subreddit, I will try to remain on track.
The theological deliberations of the time focused on the perceived prevalence of sins as the cause of the plague. They believed, and this is probably the single most espoused fact about the Black Death, that the plague was a divine punishment for sinful behaviour. One of the most common responses of the clergy, was to stress the importance of praying and penitential activity.
Regarding your specific inquiry on reactions and strategies, David Nirenberg interestingly notes that parts of the Crown of Aragon cracked down upon gambling and other sinful activity. Noticeably this was in anticipation of the plague, not during it. Nirenberg believes that these measures were put in place in order to dampen the onslaught of the plague, essentially by lowering the amount of sinful activity in circulation. For comprehension, this is akin to what we would call a prophylaxis in modern medicine. Certain sermons and masses were also held following this very idea.
To give you an idea, here is William Edenton, Bishop of Winchester, writing to his diocese in 1348, also trying to minimise damage caused by the plague (page 117 of Horrox 1994):
We also order that every Friday you should go solemnly in procession through the marketplace at Winchester, singing these psalms and the great litany instituted by the fathers of the church for use against the pestilence and performing other exercises of devotion, together with the clergy and people of the city, whom we wish to be summoned to attend. They are to accompany the procession with bowed heads and bare feet, fasting, with a pious heart and lamenting their sins (all idle chatter entirely set aside), and as they go they are to say devoutly, as many times as possible, the Lord's Prayer and the Hail Mary. They are to remain in earnest prayer until the end of the mass which we wish you to celebrate in your church at the end of each procession, trusting that if they persevere in their devotions with faith rectitude and firm trust in the omnipotence and mercy of the Saviour they will soon receive a remedy and timely help from heaven.
There were also the flagellants (as I'm sure you've heard of) and other millenarian movements. They were not really considered part of 'the church', so it doesn't relate directly to your question, but they were nonetheless 'popular' movements based on millenarian theology (the idea of an imminent change - typically caused by the coming of anti-Christ, Christ, the Kingdom of Christ and the Last Judgment in various configurations) and fervent penitential activity.
'Scientific explanations' is Horrox's second category of responses. Many of these came from Paris, home of the leading faculty of medicinal studies of the time. Some of them believed that harmful constellations of celestial objects led to the creation of the plague. Others took an approach based on the idea of miasma (a theory of air pollution), or argued that the plague was caused by a bad composition of air. This is what I know least about, but from your question I gather that this is not part of your main inquiry anyway. What's important to note here (in order to avoid a presentist interpretation of these explanations as desperate 'pseudo-science') is that these explanations were the at the forefront of scientific inquiry and reason. In addition, these explanations were fully compatible with the theological explanations presented above.
Horrox's last category of responses is what she calls 'human agency' - responses containing the idea that a conspiracy was behind engineering and spreading the plague. Common targets of suspicion in this period were usually Jews, lepers, pilgrims and the poor. But mainly the Jews (and I will have to strain myself not to digress here). In response to the plague, persecution of Jews started to emerge in many cities - particularly in the Rhineland, in which widespread mass executions and collective burnings of Jews became the norm. Some believed that the plague could be stopped by killing the Jewish conspirators. But when plague eventually returned to the cities later, it became obvious that the Jews were not to blame anyway.
In response to these widespread persecutions of Jews accused of disseminating the plague by food and water, Pope Clement VI had something to say. He issued two Papal bulls in 1348. The first bull was a reiteration of the formal papal stance of protection of Jews, otherwise known as the Sicut Judaeis. Another bull (which was actually issued twice), the Quamvis perfidiam Judeorum, gave further commentary on the idea of a Jewish conspiracy. (The bulls can be found in the Horrox source book and in John Aberth: The Black Death (2005)). The pope denied that the plague could have been caused by Jews. To put it simply, he believed it would have been beyond their power. Rather, he seemed to believe that something as calamitous as the plague, must have been the caused by greater things than mere human agency (thus alluding to the two other types of explanations presented above).
Summing up what I've said:
These are some good books:
Also check out the BBC podcast In Our Time on the Black Death for a nice overview by some pretty prolific scholars.
Follow-up question: How wide-spread was the plague? Did it reach islands such as Iceland? Were the Russians affected? What about the Middle-East, etc.