Arduino AC Motor Control and Automation?
Single-phase motors are not a good choice for speed control. Triac controllers control the motor voltage. That works reasonably well if the motor is driving a fan or centrifugal pump. Even with fans and pumps, some motor designs are not suitable. With that type of load, the required torque decreases a lot as the speed is reduced. Water or air flow and motor efficiency decreases similarly. There is not much point in operating below 1/3 or 1/4 speed.

For AC motors, it is much better to control frequency. However variable frequency drives (VFDs) don't work particularly well with single-phase motors.

Brushless, permanent-magnet motors are best for very low power requirements.

Choosing between three-phase induction motors and brushless permanent-magnet motors is not easy. They both can provide the performance required for practically any application. Permanent magnet motors are generally more efficient. The price depends on all of the technical requirement details, quantity, location etc.

Triac controllers may require a little work to interface with an Arduino, most others will be easier.