As a FI instructor I see an almost universally poor understanding of even the simple systems found on light aircraft.
You should be very familiar with everything in the POH of the aircraft you are going to fly. A test I use on my new FI students is to pull out the electrical and fuel system schematics and get them to explain to me what all the lines and squiggles mean.
You need to be
completely familiar with all the emergency procedures and more importantly should be able to give a short accurate description on
why each of the checklist items are being carried out.
On a more general topic one of my pet peeves is the lack of evidence based training. Accidents by low hour pilots usually have a basis in poor training.
A study of the accident record reveals a pattern of a relatively small number of repeating accidents. Broken off nose wheel/prop strikes caused by repeated nose wheel first touchdowns is a good example of a common, but totally preventable accident. As you get into your training ask your instructor how to ensure your students don't continue the cycle.
Finally the European training system seems to have a fixation on teaching aeronautical theory to sometimes absurd lengths and at the expense of practical information.
Theory instruction during actual flight training ( Not ground school ) should IMO be prioritized as follows:
Must Know = Theory whose understanding directly impacts flight operations (eg relationship of temp and humidity to the formation of carb ice, why wing tip vorticies are dangerous)
Good to know = Theory has an indirect impact on flight operations ( eg the relationship between AOA and lift, how clouds form)
Nice to know = Theory which will never be used on a typical flight ( pretty much everything on the written exam
)