Did anybody learn anything?

Definitely. Many of the things we now consider basic for managing a pandemic were realised during the Black Death, though much was forgotten in the decades that followed.

Starting in 1347 with the Black Death's arrival, it was noted very quickly that the plague spread from person to person. The idea of contagion from human to human contact may seem very obvious to us now, but at the time many believed that disease was sent by God, and they lacked the technology to know for sure what vectors the disease took. The Black Death was particularly difficult in this regard, because it actually spreads in two ways: fleas and droplets. One of the things we see in surviving records from the years of the pandemic itself is a battle between religious institutions, which favoured the 'sent by God' idea, and urban administrators who generally (though definitely not always) favoured contagion theory even though there were many competing theories about how the disease actually spread.

When people realised in the years afterward that plague was not a one time problem, but a disease that would keep coming back, many cities decided to do something about it by appointing civil servants to positions of responsibility. In some towns they were known as 'Commisars of Public Health', or my personal favourite, 'Plague Guards'. Many towns set up not just one official but an entire government department, equivalent in role to modern health boards like the CDC (though nowhere near as sophisticated, obviously). Sometimes this work was rolled into another job - for example the Italian town of Lucca placed this responsibility with the saggior officiale delle vie e de’ pubblici (senior official of the roads and public works). In Lucca's case, we can see some concern about sanitation - Lucca saw disease prevention as a matter of infrastructure, such as better sewers. The job of these officials could vary, but they shared the common purpose of watching for the plague's return and assisting the leadership in their decision making. One of the main legacies of the Black Death was the idea that public health is actually a thing, and one that should have civil servants dedicated to it. That alone was a big step forward, though much of it was forgotten in the early modern period as plague became less of an issue. But when we turn on the TV today and see people like Dr Fauci advising politicians on coronavirus, that role is a descendant of the so-called 'Plague Guards'.

As to what these Plague Guards were advising, that depended on their own theories on disease prevention. In the wake of the Black Death, many treatises were written on the subject, mainly in Italy. A common theme was the idea of quarantine, and what we currently refer to as lockdown. These were everywhere - we find them in responses to disease outbreaks in Sardinia, Germany, the Balkans, France, Italy, though not so much in England for reasons we aren't sure of. If plague showed up in the neighbouring town, the gates of your own would often be shut in response. In late medieval Germany, the favoured method was to make people swear they hadn't been to an infected place (and preferably prove it), and to leave their goods outside the city walls in the open air for a few days to let the 'bad air' commonly blamed for disease leave the items. Many towns facing a plague outbreak also banned public gatherings, with some towns such as Tournai practising this method during the Black Death itself. Initially the town attempted measures like curfews and limited numbers at funerals and weddings etc, and as the Black Death kept getting worse they even banned the ringing of church bells just in case they caused townsfolk to gather. Some building blocks of minimally competent pandemic response - limit public gatherings, quarantine new arrivals, making sure someone actually smart helps make decisions - were learned during the Black Death, although many were forgotten afterward as plague became less common and the need to know this stuff became less important.

Another effort we see after the Black Death is investment in public health infrastructure and the training of physicians. In some regions, especially Italy, medical training was increasingly taught in universities and controlled by guilds. Prior to the Black Death, anyone could call themselves a doctor, but afterwards there was more interest in regulating and encouraging the formal training of medical professionals. Accompanying this was an expansion of medical facilities such as sickhouses (literally just places to dump sick people so they didn't spread their illness to everyone else) to public hospitals that could offer what passed for medical care back then. There were also increased public sanitation regulations. For example, in 1350 just after the Black Death had passed over London, the city banned bathing in common waterways like the moat of the Tower of London or the Thames. We don't know much about London during the Black Death (record-keeping seems to have partially collapsed), but from regulations like this we can see that some attempt was made to identify a cause and limit the damage plague could do. Some Londoners probably thought it spread through the water supply, so they banned communal bathing. A flurry of public health reforms show up in London's records in the decade or so following the Black Death, and from court records we can see that they were enforced pretty well, but in the 15th century these regulations increasingly went unenforced and lessons were forgotten.

That being said, many of the things learned in the wake of the Black Death are actually pretty hard to tie directly to it. We don't see a clear pattern in late medieval records of plague outbreaks leading immediately to public health reform beyond short-lived bursts of legislation in some places. What we see instead is an increased awareness of public health more generally and a greater willingness to invest in the tools to fight it. For some local governments, this meant investing in the training of physicians and the establishment of public health boards. For others, it just meant some minor changes to sewage and bathing regulations. However, it's worth noting that many of these lessons were forgotten as the Black Death itself was gradually forgotten too and the urban population growth of the early modern period made most late medieval health reforms unfit for purpose.

Sources:

Carmichael, Ann G. "Contagion theory and contagion practice in fifteenth-century Milan." Renaissance Quarterly 44.2 (1991): 213-256.

Ciecieznski, Nathalie J. "The stench of disease: public health and the environment in late-medieval English towns and cities." Health, Culture and Society 4.1 (2013): 91-104.

Coomans, Janna. "The king of dirt: public health and sanitation in late medieval Ghent." Urban History 46.1 (2019): 82-105.

Geltner, Guy. "Healthscaping a medieval city: Lucca's Curia viarum and the future of public health history." Urban History 40.3 (2013): 395-415.

Kinzelbach, Annemarie. "Infection, contagion, and public health in late medieval and early modern German imperial towns." Journal of the history of medicine and allied sciences 61.3 (2006): 369-389.