It is not so critical to memorize what to do about diseases, pests, and catastrophic hive events (you can always look that up).
It is critical to recognize them!
That ability to recognize a disease process in progress is arguably the most important beekeeping skill: here then is a quick summary of what should set off your beekeeperly alarm bells:
Survey the Open Brood
All, all, ALL!!! of the open brood should be fat, juicy, pearly white, gleaming and wet looking. All of it. If you see any brood that are dried up looking, off-colour, in a strange position, or which look rotted or unnatural, pull the alarm bell.
European Foulbrood, caused by the Melissococcus plutonius bacterium affects open brood. Brood before capping. It is transmitted from infected nurse bees by mouth as they feed the larvae their gut contents. Over time the hive’s stores of nectar and bee bread (the fermented pollen fed to larvae) will be infected as well, passing through so many guts and infecting new bees.
The cappings on the brood should look solid and intact. American Foulbrood, caused by the bacterium Paenibacillus larvae ssp. larvae generally kills the bee larvae post capping. The bees sense that the larvae under capping is dead or dying and they will uncap ragged holes in the brood, in their attempt to remove the problem larva. Unfortunately, removal is by mouth and this infects a new cohort of bees, who in turn infect all their fellows and the new larvae they feed.
Both the Foulbroods will, early or late, smell nasty. Like rotting meat or smelly old socks in a gym bag. Bad smells in the hive are always an indication that something is amiss.
Treatment is a combination of medication and getting the bees off the old, contaminated comb and hive contents. In the case of AFB, resistant spores are generated that can only be destroyed via irradiation (ie. at an Iotron facility) or burning.
Other brood diseases (sacbrood, chalkbrood) all exhibit deformed, off colour, odd or dead larvae. If you see something unusual in your open brood, or notice that developed bees are remaining in the cells, unhatched and dead, identify the cause immediately and take remedial action ASAP.
Examine the Brood Pattern
A healthy, vigourous queen should lay solid sheets of brood. If they are spotty “shotgun” patterns, examine carefully for disease. Shotgun brood pattern can also be from a poor queen, a queen infected with Nosema, or Varroasis. Sometimes a new queen or new nuc may display a shotgun pattern, but within a couple of weeks you should see solid sheets of brood. If the shotgun pattern persists, pull the alarm bell!
Also check that you have new eggs in the hive. When you do not, find that queen. Is she missing? Or is she not laying? Why isn’t she laying? Maybe a swarm is about to go off. Maybe she failed to mate successfully and is infertile. Figure it out.
Check the brood cappings. Is most of the capped brood smooth worker brood? Or is your queen laying only raised drone brood, indicating she failed to mate (fertilized eggs make worker bees, unfertilized eggs make drone bees). An infertile queen needs to be replaced. Note that her hive cannot raise a replacement queen from drone eggs!
Examine the Adult Bees
Are your bees all plump, normal body proportions, bustling happily about? Or do you see deformed wings (indicating Deformed Wing Virus, vectored by Varroa mites), stumpy bodies, tiny wizened bees, or black, greasy bees?? Are they wandering about looking dazed, unable to fly (possible tracheal mite infection), or are they trembling constantly (possible Israeli Paralysis virus, also mite-vectored)?
Always be on top of the Varroa mite situation. Not only are they deadly on their own, unchecked, but they vector a host of diseases.
If the adults look unusual or wrong, then something probably is wrong! Get to the bottom of the issue immediately.
Examine the Hive
Do you see fecal streaking on the exterior of the hive, or on the interior? That is abnormal and indicates temporary dysentery at best (contaminated/fermented feed somewhere??) and pesticide poisoning or Nosema apis at worst. You need to medicate for Nosema immediately and hope that works.
Is there mold and/or dead bees in the combs, and/or wax moths? If there is, reduce the hive size to make sure “the colony fits the hive” ie. they have a home size they can keep clean, warm and patrolled. Make sure they are healthy, and that the hive is dry and ventilated.
Remember that strong hives, particularly in a nectar dearth, will attack weak hives, stealing their honey. Reduced entrances and robbing screens are the answer here. If robbing is in progress, completely shut the hive being robbed, make necessary changes and hope the robber bees do not return next day. Move the robbed hive if necessary.
Wasps will rob out hives, eating eggs, larvae, adults (all good protein sources for baby wasps) as well as the nectar and honey. This is common in late summer, when wasp populations are at their peak and raising queens to overwinter.
Be careful, particularly in a nectar dearth, not to spill feed or honey in the beeyard, which can set off a robbing frenzy, or feed from open feeders (ditto, and also a disease transfer station!), or leave honey filled hives open (invites hungry robbers in). Do not leave undefended honey supers in the beeyard: they will be robbed in a twinkling.
If a hive has been robbed, check the next day (when things have settled down) to ensure the queen was not killed in the raid.
Evaluate the Vigour of the Colony
Always ask yourself if the hive is as vigourous as it should be for the time of year. Is the queen laying properly, is the hive building in spring and remaining as strong as it was a month ago? Or is it stalling out and/or dwindling for some reason. It could be you have a failing queen, or Nosema ceranae loose in the hive. You could have a sick hive, or one being predated by skunks at night (they love to eat bees!). Maybe it’s honey bound and there is no room for your queen to lay.
Any colony that is not bringing in pollen when others are, is failing to store nectar and honey, or is quiet at the door (little bee traffic, particularly when nearby hives have lots), or is failing to take a syrup feed (especially if other hives are taking it well) has a problem.
If you colony is not doing well, there has to be a reason. Find it.
There are an abundance of books and web based information pages to help you identify and remediate your hive issues. It also helps to join a good local bee club, and find a few older more experienced beekeepers as mentors. Take a bee course, as that guarantees an instant mentor! Go online and ask questions (include photos if possible) on bee forums. There is lots and lots of support out there.
Note that in the USA, several universities offer online courses leading to the Master Beekeeper certification.