Best way to control this AC motor
The motor is probably a single-phase induction motor. Induction motors can not be controlled very well by controlling the voltage. Induction motors try to run at their synchronous speed regardless of voltage. Reducing the voltage reduces the amount of torque that the motor can produce, makes it weaker. That tends to make the motor slow down. If the motor is driving a fan or centrifugal pump, considerably less torque is required to run at slower speeds, so the motor can slow down a little and continue to run ok. If the load requires the same torque at any speed, the motor is likely to slow down only a little and then stall and draw high current. Even a fan or pump might not operate at less than 1/2 to 3/4 speed.

You can buy a variable frequency drive (VFD) for a single-phase induction motor, but few manufacturer's make them and they tend to be rather expensive. They are better than voltage reduction devices, but they are not recommended for any type of load except fans and pumps. They also can not be used with capacitor-start motors, only shaded pole or capacitor-run motors.