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My final thought was just rereading Learning Perl again and again until I had the whole book memorized. That sounds pretty boring and I'm sure many will agree with me on that.
How many perl program did you create in the course of reading Learning Perl? Did you put them in version control?
I suggest next, you read http://learn.perl.org/books/beginning-perl/, but this time, do each of the (~3 on avg) exercises listed at the end of chapters 2,3,4,5,6,7,8,11
After each chapter, after doing the exercises, write a short review, compare it to a similar chapter in Learning Perl. Try to figure out if you learned something new, how you liked the way the information was presented, was the pace better, did you have to unlearn something...
Next I would go to Perl Exercises: For Beginners :) and first do the .edu computer science courses exercises, then some project euler exercises (say first 10)
Then I would take a long break, then come back and review your progress, and review some concepts that gave you trouble during the exercises
Then i would do one or two of the katas -- and put each part into version control
At this point, I would read through Modern Perl, and explore the modern ways to solve some of the exercises you've done so far -- this is where having lots and lots of little programs in version control helps you :)
Next I would start solving real problem , and maybe do another euler exercise and a kata in between :)