There are a few nodes here at The Monastery that you'll be able to find ideas on where to start (i.e. what type of projects may help motivate you to learn perl in your own way) - use Super Search. Also, you may want to check out the book Learning Perl and the Tutorials here.
In addition to the suggestions by moonie, you might
consider getting a copy of Perl Cookbook. Browse
through it and try playing with the examples given.
I'm sure you'll find lots of examples on topics that you
are interested in. All examples are given in a
Problem/Solution/Discussion format, are relatively short
and will likely address most of the basic problems you
will encounter as you begin programming Perl.
Moonie is right, this is one of the best place to start. Register and have some fun with us. Also, Learning perl is a wonderful book, each chapter covers a principle of perl and is followed by exercises to reinforce what you have learned.
Moonie there mentioned the Tutorials section of perlmonks, but the Library is even more useful. Besides containing perl pod, it also contains many exercises.
The other place to look for ideas is to start reinventing little wheels, like a visitor counter ... check out merlyns website for ideas http://stonehenge.com/merlyn.
That should be more than plenty to get you doing something ;)