My experience has been that SysAdmins generally approach Perl to solve daily task automation problems. Perl is great for this. Perl also allows for so much more than that.

The basics that everyone who uses Perl more that casual dabbling should know are libraries and modules for code reuse and algorithm standardization. Once you know how to work well with these, you can explore OOP. OOP is more than just package creation for the sake of package creation. OOP gives you rich inheritance capabilities and allows for simplified designs and interfaces which can easily fit into much larger projects. I've seen good OO and I've seen bad OO. When you work with good OO it can make bad OO seem really bad. Of course, this is not to say that everything should be done in OO. Perl is not Java. :-)

That said, learn the advantages and disadvantages of OO design. This subject trancends Perl programming and goes into general programming practices.

Ultimately, what you need to know will really depend on what you want to do. Do you want to head into the world of web programming? You will need to know various templating technologies. Do you want to work on backend data-driven applications? DBI is your friend (and occasional enemy). Define your realm and the tools you need to learn will become apparent.

Good luck with your pursuit into Perl programming. Remember to always think "How can I generalize this just a bit more so that I can re-use it over there, too?" That question can be the difference between enjoying your product and hating it.