The "Perl Cookbook" and "Perl by Example" really helped get me started. If you buy them, make sure to buy the far cheaper used ones. I got both books for under $10!
I do however hold that these books are only useful for getting you started and teaching you the basics. To actually become proficient with Perl, you have to tackle projects. This could literally be anything; just come up with an idea and do it. For example, one of the first things I did was create a script for an NFL weekly Pick'Em league that takes a .txt list of the current week's winning teams, and then highlights the correct picks in a spreadsheet. You could start even smaller, like just modifying a text file, and build up from there. I'd also suggest just looking for modules that seem useful/powerful and tinkering with them to see what you can do.