"Single phase asynchronous" is a less common way to describe an
induction motor. The type in the picture is a run-capacitor, split-phase motor.
Where does this diagram start? With the red wire to the far left?
The diagram only shows a motor, so it doesn't really "start" anywhere. Normally power enters on the black and white wires; this is shown by the ~ symbol for AC. Black and white are standard colors for Live and Neutral in North America.
Why isn't there a motor icon with all the wires branching out as a start point?
A simpler device such as a DC motor can be described with a "M in circle" symbol. This device is more complex.
Inductors are required? ugh oh. I don't have them anymore. What are "M", "A"?
No. The "inductors" in the diagram are the motor stator windings. These motors have a Main and Auxiliary winding, the Auxiliary is used for increased starting torque.
There are two red wires I don't know the difference.
Since they are used to connect the (non-polarized) run capacitor, they are interchangeable for this use.
What is that symbol that looks like an open switch with an extra symbol on top?
A thermal cutout or fuse. They are normally a small metal cylinder with axial leads.
Can I use a potentiometer (or something better) to easily vary the speed?
To have varying speed it would be easier to use a DC motor with a rheostat. This induction motor turns at 1800 rpm idle and less under load.
The formula for induction motor speed is \$ RPM = 120 \cdot \frac{f_AC n_poles} \$
You can alter the speed by changing the phase between the Main and Auxiliary windings, by changing the capacitance attached to the colored wires. Variable capacitors are usually implemented using electronic circuits.
I saw in a youtube video resistance can be tested on each wire to find which ones control the corresponding speed winding. Do I need to do that?
The windings are already depicted on the diagram.