As everyone has said working through the books and example exercises is the best place to start. As you work through the examples try to relate them to something you are familiar with. If you can try to relate them to how you could use them in your own area of interest, you will find them much more enjoyable and easier to grasp.
When you have the basics try creating your own simple networking scripts. You could try creating a simple ping script, open a connection and communicate with a remote server, try SNMP to one of your switches. This may seem daunting at first, but use cpan modules and look at the examples in their documentation, you will see its not that scary. Don't worry at this stage about creating something new, re-inventing the wheel is a great way to learn as you can compare your solution to others. As you create these simple scripts you should gain a better understanding of how perl handles networking tasks. This should help you start to form ideas of how to solve problems you encounter and automate simple tasks.