Pick an 'easy' problem. Say, loading a CSV file with your expenses in it and producing a short report of aggregate amounts ("I spent $500 on beer this month?!"). Then add features. Little ones.
Maybe the next step is to highlight high values. Maybe you build a simple database with DBD::SQLite2 so that you can track changes in expenses from month to month. Maybe then you dump your summary report back to a CSV file. Or display it in a Tk widget.
I found the best Perl advice I got was to port simple UNIX commands to Perl (it's been done, but doing it yourself teaches you tons!). For example, make a Perl program that does everything *NIX's ls does. You can start with the basic functionality, then add each feature slowly (bonus points for putting each feature in a module).
Exercises like this will get you familiar enough with how Perl works (even some of the quirks) that you can start on bigger things. Look for CPAN modules with known bugs or that are asking for feature enhancements. See if you can fix/add to those modules. Then you get the added advantage that the author will provide you feedback about your patch if it needs improvement. That's very much like a mentor relationship.