Connect your iPod to your computer via the USB cable. Plug one end of the cable into the port on your iPod, and plug the other end of the cable into a free USB port on the front or back of your computer. Do not use a USB port on your keyboard. You need a high speed USB port and the USB ports on a keyboard are not high speed USB ports and will not work correctly for an iPod.
Wait for iTunes to open. If it doesn’t open automatically, double-click the iTunes icon on your desktop to open it. Alternately, click the Start button in the lower left corner of the screen, click All Programs, and then click iTunes.
Find your iPod on the left side of the screen listed under Devices. Click on it one time to select it.
Click the Summary tab on the right side of the screen. The serial number for your iPod will be displayed on the screen. If you have a 1st or 2nd generation iPod Shuffle, click the Settings tab instead of the Summary tab. It will be listed in the information in the iPod section of the screen.
If you don't have access to iTunes, you can still find it in other ways.
The iPod About Screen
You can also find the serial number for your iPod on the About screen. To do this for the iPod Touch, tap the Settings icon on the display screen, tap General, and then tap About. The number will be listed on the screen. For all other iPod models, select Settings, and then select About. Press the Center button until you see the serial appear on the screen.
On the iPod Itself
Depending upon what type of iPod you have, the serial number is either engraved on the back case, or is on or below the clip.
Where is the serial number on an iPod Touch? On both the regular iPod and the iPod Touch, it is engraved on the back case towards the bottom.
On the iPod Nano (6th generation), it can be found on the inside clip edge touching the back of the device. For earlier models, it is engraved on the back case towards the bottom of the iPod.
On the iPod Shuffle (2nd generation), the serial is engraved on the inside edge of the clip on the back of the iPod. You have to open the clip to find it.
For other iPod Shuffles, you can find the serial number on the back of the iPod (the side with the Apple logo) on the bottom right side.
When reading the serial number on the back of the iPod, the letters and numbers are very small and can be hard to read for some people. You can use a magnifying glass if you need to.
Other Places to Find the Serial Number
You can also find the number on your iPod packaging. If you still have the original packaging, you can find it on the barcode label affixed to the package itself. This label is white and the serial is in black lettering.
You can also look on the invoice or receipt, as some retailers include it on the store receipt at the time of purchase. Not all retailers do this, however.
You can also look online on Apple's website at your My Support Profile. If you used your Apple ID to register or purchase your iPod (available only in the United States and Canada), you may find it online. Log in using your Apple ID and find your iPod in the My Products box on the main page. The serial will be listed there. If you have more than one Apple ID, they will all be listed here. Be sure to select the correct Apple ID that goes with the iPod you currently own.
When You Need Your Serial Number
When you first purchase a new iPod, during the set up process in iTunes it will ask you for your serial number. If you have problems with your iPod in the future, you can enter this number on Apple’s Service and Support website, and they can check to see if it is still under warranty. If your iPod is ever stolen, the police will want it for identification purposes. In some cases, you might find your iPod if someone connects the iPod to another computer and enters the same number. Once you find the number, you may want to write it down in a safe place to have for future use.
Source and Image Credit: https://support.apple.com/kb/HT1459