What happens if only rudder is applied in a turn without ailerons? Do the two have to be applied together all the time?
To turn – any vehicle, not just an airplane – you need to generate corresponding centripetal force.

When you only apply rudder, the plane will turn a bit, but because unlike boat it has no keel, it will not generate much centripetal force, just fly somewhat sideways, which is called a skid. It will produce more drag this way.

Now any well designed plane has yaw-roll coupling, which will cause it to bank into the turn anyway. It is caused by the slight increase in speed of the outside wing, and the dihedral most wings have resulting in increase in angle of attack, thereby increasing lift on the outside wing. So you can make the plane turn with just the rudder, but it will be inefficient, and it will be slower as the yaw-roll coupling will take its time.

On the other hand if you apply just the aileron, the plane will bank into the turn as usual. However, there will be some slip, as the deflected ailerons cause an adverse yaw – the aileron deflected down, which is on the outside wing, causes more drag – so the plane will actually turn a little less than it should. Like skid, that will cause extra drag.

What you want to do is bank into the turn with ailerons, and use rudder to compensate the various drag asymmetries to keep the slip-skid ball centered. That will give you least drag. Different planes will need slightly different application of rudder.



There are cases where you will want to apply ailerons and rudder in different combinations (as mentioned in comments below):


On the ground you steer with rudder, while using aileron to prevent wind from tipping the plane sideways – the landing gear provides the centripetal force in this case.
During cross-wind landing, you need to align the plane with the runway, but as it flies relative to the air, it tends to drift sideways. You need to make it fly sideways relative to the air instead by banking into the wind while using opposite rudder to prevent turning.
The same combination applies during cross-wind take off, aileron into the wind to prevent wind from tipping you over sideways and rudder to compensate for weathervaning.
You can also use bank with opposite rudder to increase drag if you are too high and fast on the approach – this is called forward slip. Just don't let the airspeed decay too much, because stall in slip or skid is more likely to result in a spin.