That is actually a very good question and something that Boeing has developed a system for. They call it EICAS (Engine Indicating and Crew Alerting System), other manufacturers have a similar system but have a different name.
It monitors a lot of parameters and sensors (A LOT). If it detects a 'major' problem like say an engine fire, it will alert us both aurally and visually. Just about everything non-normal is shown in text on the middle screen (the one with the two rows of white lines in the image) in either red or amber.
It is smart enough to not overload us with information; If you have an engine failure you will also have low oil pressure for instance, but that warning will be inhibited.
On the model pictured above (the 747-8) we have an electronic checklist; We can push the 'checklist' button and it will automatically show the 'engine fire' checklist on the screen in this example (below the one in the center with the white lines).
Is it foolproof? No, we still have to keep thinking about what's going on and confirm the problem if we can.
The system is designed for singe failures. Multiple failures at the same time and physical damage may throw it off. The same with the checklists; Even though two failures are correctly identified, it can depend on the situation which checklist you want to start with, it doesn't have to be the one the the airplane comes up with first. (It may tell you to shut off something that you need in the next checklist for instance).