If you are studying for the JAR ATPL(A) exams then the things that you need to know are:
Vx is the speed at which maximum angle of climb is greatest.
The Sine of the climb angle = (Thrust-Drag) / Weight.
The Sine of an angle increases as the angle increases up to 1 at 90 degrees. So if we increase the Sine we will increase the climb angle. This means that for maximum climb angle we need maximum (Thrust-Drag) / Weight.
For any given weight the climb angle will be greatest when Thrust - Drag is greatest. Thrust-Drag is called excess thrust.
For jets, the thrust against TAS line is assumed (For POF purposes) to be horizontal, so the excess thrust is greatest at the speed at which drag is minimum. This speed is Vmd. So Vx jet is Vmd.
But for props the thrust line falls off rapidly with increasing airspeed. By the time we get to Vmd we have lost a significant part of the thrust. So the greatest excess thrust does not occur at Vmd. For the typical JAR ATPL(A) prop aircarft, excess thrust is slightly (perhaps a knot or two) less than Vmp. So Vx prop is slightly less than Vmp.
Vy is the speed at which the maximum rate of climb is greatest.
Rate of climb = (Power available - Power required) / Weight.
Power available - Power required is called Excess Power. So we get best rate of climb at the speed at excess power is greatest.
Power required is equal to drag x TAS. This means that as speed increases we are multilpying a bucket shaped drag curve by a straight line increase in TAS. This converts the bucket shaped drag curve into a power required curve that looks rather like a Nike Tick.
Power available = Thrust x TAS.
When TAS is zero, the power available = thrust x zero, so power available is zero when TAS is zero.
In jets, the constant thrust multilpied by a linear increase in TAS produces a straight line power available curve sloping up to the right as TAS increases. If we plot this on the same graph as the Nike Tick shaped power required curve we see that the greatest difference between the two (the excess power) occurs at a speed higher than Vmd. This is typically around 1.32 Vmd at low altitude. So Vy jet is greater than Vmd.
In props the thrust reduces as TAS increases. So when the downward curving thrust line is multiplied by a linear increase in TAS we get a power available curve that is bit like an inverted Nike Tick. This starts at zero when TAS is zero, increases with increasing TAS until it reaches a maximum value, then decreasing with further increases in TAS.
If we plot this on the same graph as the power required curve we will find that maximum excess power occurs at a speed slightly (a knot or two) higher than Vmp. So Vy prop is slightly higher than Vmp.
All of the above is of course just text book theory, so real-world aircraft will behave slightly differently.
We can find the speed for lowest sink rate (rate of descent) in a glide by looking at the energy situation.
When the engines are operating they give the aircraft energy. This is constantly being used up to do work in pushing the aircraft forward through the air. If the engines fail, then the aircraft can get no more energy, so it must use whatever energy it posesses at the time of the failure.
At any point in a flight an aircraft posesses kinetic energy (1/2mVsquared) by virtue of its velocity, and potential energy (Wh), by virtue of its height above the ground. When all of this energy has been used up the aircraft will have no velocity and no height, so it will be be standing still on the ground.
Glide endurance is the time that is taken to use up all of the energy posessed by the aircraft. The lower the rate of energy consumption, the longer will be the glide endurance. But the rate at which energy is consumed is power. So for minimum energy consumption rate and maximum glide endurance, we must fly at the speed at which power required is minimum. This speed is Vmp.
Glide endurance is also equal to height divided by sink rate. So when we have maximum glide endurance we also have minimum rate of descent. So Vmp can also be described as the speed for minimum sink rate.