Here's a suggestion if you're interested in web-related perl projects: there was a nice intro to Ruby a couple of years ago here:
Rolling with Ruby on Rails
Even though this article is on Ruby, it's very well written and it introduces a good sample problem: build an online cookbook (i.e., an online recipe database). You want to be able to create new recipes, modify old ones, view them, search them, etc.
While it seems simple, by the time you work through it you'll have a good understanding of perl, as well as mysql, apache, linux (the whole LAMP stack).
Now, the point of the original article was that you could do this in about 20 lines of ruby... nevermind all that!
You can do something comparable using Catalyst. But as a beginner, I suggest doing it the hard way with CGI::Application, because it will help you understand all the nitty-gritty details. You can always do a "version 2" in Catalyst later. That will really help you understand and appreciate what all a web framework does for you "under the hood".