It's an AC motor I presume? You should onyl use a TRIAC with an AC motor.
TRIAC - switching speed control
relay - just on or off for continuous periods of time
Tr. = transformer? That's only needed if your source voltage is different from the motor voltage (and you would use it in addition to a switching device).
Regarding the switching speed control, it's not as simple like PWM with DC motors. I don't know much about variable speed AC drives so...Most AC motors are designed for constant speed operation.
You can control an AC motor with a triac but for a DC motor you need to use a transistor, bipolar or mosfet. A little more detail on what you want to do would help.
Yes, a properly rated TRIAC should work just fine. I don't think there is a problem with using a TRIAC that can handle 220V in a 24V circuit as long as you can meet the switching requirements in the same way you can use a 800V FET in a 24V circuit.