Pilots don’t always seem to know where to look when something has gone awry. When an engine begins to vibrate we should quickly determine if it is a major vibration and, if so, shut that engine down quickly: you wouldn't want it shaking itself off it's mounts, or anything that comes after. If it’s not that bad may we should look at the engine’s power indicators (i.e. Manifold Pressure, RPM and Fuel Flow) for a piston powered engine. Then we can look at the condition indicators (EGTs and/or TITs, CHTs, oil pressure, oil temperature, etc).

Look at the cowling for oil streaming back. It doesn’t take a large loss of oil to make a big mess. If the oil pressure hasn’t started to drop you may be able to reduce power and continue flying to a good landing at an acceptable site – preferably an airport. However, what about the flight instruments, what do we look at first? In almost all cases the airspeed is the first indicator to look at. If you haven’t changed power settings and the airspeed is less than before, possibly the pitot tube is icing over, there is induction icing,  the airplane has begun a climb, or you’re in a downdraft and the autopilot (in order to maintain altitude) is pitching the nose of the aircraft up. You could have even bumped the power lever aft unintentionally.  If the aircraft’s heading is not what you’ve selected, or the aircraft is climb/descending when it shouldn’t be, it is possible the autopilot has failed, or that you could have accidentally disengaged it.

The main point is: I see pilots not looking at the right thing when things go the wrong way. They ask the wrong questions and look at the wrong things for answers so they don't get the answers they need, and end up making the wrong decisions.

Also, to solve the age old question: In order for a pilot to climb/descend at a specific airspeed and specific rate, should he use power for airspeed and pitch for rate? Sure. How about power for rate and pitch for airspeed? He can do that too. However, I would personally use pitch for whatever is most critical (or whichever you’re already closest to) since the airplane responds quicker to pitch inputs as opposed to power inputs. 

We've learned that pilots can look the wrong way because they ask the wrong questions, so then, what are the right questions?