So, for someone who has never read or written Perl code, where should I start? Perl 5 or 6? I get that they're different languages, but I want to see what "Perl" is like.
I'm not sure you do get it, unless you are talking about the "feel" of using it, in which case use either. They aren't just different languages because there's some things that break backwards compatibility, they are vastly different in many ways.
If you want to know what current deployed Perl is like, look at Perl 5. If you want to look at what the "Perl" future is, look at both Perl 5 and Perl 6, since they'll both be around. If you want to see something new and different, look at Perl 6.
I was talking about the "feel", indeed. So for example, if someone wanted to experience what a Lisp is like, I could point them to Scheme.
I want to know what "Perl" is like, but the fact that Perl 5 and 6 share the same name is a tad confusing for a newcomer. Precisely because I have no idea what "standard" Perl is.
I guess Perl 5 it is, since it's already on my machine.
Then I recommend perlbrew[1] for Perl 5, and rakudobrew[2] for Perl 6 if you decide you want to take a look there as well. They will let you quickly build and install any particular version of Perl you like, and allow switching between versions if you have the need. The Version of Perl included in most distros is rather old as, Perl 5 sees regular releases with new features.
The one main hint I have for you (beyond reviewing the Modern Perl book, as my sibling comment recommends) is to understand context and what that means in Perl. Perl can look deceptively similar to C and other procedural languages, but there are some somewhat (still) radical ideas underneath that will leave you either scratching your head or grumbling in frustration if you don't understand context (list or scalar), and how it's relevant in almost everything Perl does.