How do you increase and decrease the speed of an AC motor?

You really can't really "control" the speed of a the commonly available home-use 120 VAC fans.

The problem is that most of the motors on the "regular" fans are synchronous motors, and they want to sync with the line by virtue of frequency. Fans with different speeds have different windings in them that are specifically set up for the different speeds. A switch allows the different windings to be selected. Simply reducing the line voltage will only reduce the torque the motor will supply, but will not directly affect its speed. The lower torques will cause the motor to slow down, and that's true, but this is not necessarily a good thing as the fan was not designed to run like this.

Sophisticated electronic variable frequency drives (VFDs) are designed to provide AC power but at a frequency you can control. Your ability to control the frequency gives you the ability to control the speed of the fan directly. But these units cost. For the smaller non-commercial units, it is smarter (and cheaper) to just get another fan with multiple speeds than to try to fool around and control the speed by some "gizmo" or something else.

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I beg to differ. Most home fans are inexpensive single phase AC motors, either with a capacitor (PSC) or shaded pole. These are not synchronous motors. A simple light switch dimmer (aka phase controller or triac controller) can be used to control the fan. Caution: be careful to avoid fan stalling when setting at low speeds. The dimmer controls the speed by reducing the effective voltage to the fan. At low voltages (and resulting speeds) the fan torque is also reduced. Thus the likelihood of stalling is higher. If the fan stalls with the power on, the motor will overheat and possibly catch on fire.

VFDs are definitely not the right choice for a simple fan. These require motors (almost always 3 phase) designed for use with VFDs (aka, inverters) or a special output filter for the controller. Otherwise the bearings will pit within months to a year. Besides, VFDs with a single phase output is very rare.