You may need your iPod touch serial number when you connect for first time with apple support for any products of them. You can find serial number in different ways. Here I describe some of the ways to find iPod Touch serial number. Usually you can turn off your iPod Touch and remove the battery. Then read the serial number that is shown as a string of letter and number combination.
You can find your iPod touch serial number in the about screen on your iPod Touch. Choose Setting > General > About. For other iPod models choose Settings > About and press the Center button until you see the serial number. You also find your iPod Touch serial number on the back case towards the bottom like the image below.