So do I wire these appliances directly to the arduino board
No. The voltage requirements for the light are too high for the Arduino to supply. The current requirements for the motor, and the voltage, are too high for the Arduino to supply.
do I have to go to relays etc.
You could use a relay for the light, but you can't PWM a relay. You can PWM a transistor, such as a MOSFET, to control the speed of the motor, but typically an H-bridge is used so you can control both speed and direction. Motor shields are pretty common.
Then How do I teach the board what I want it to do ?
Long lectures. The Arduino tends to fall asleep during lectures, though, so be sure to wake it up often.