Here's the short list of modules I don't like to be caught without. These are the ones that I don't just find convenient from time to time but rely on constantly.

• CPAN. Perl isn't properly installed until this is set up, as far as I'm concerned.
• DateTime (I might die without this one.) Also certain DateTime::Format modules to go with it.
• DBI and one of the DBD modules. (I've been using DBD::mysql, but one of the others will probably do just as well. You want the corresponding DateTime::Format module, e.g., DateTime::Format::MySQL, to go with it.) I also have my own little wrapper module with functions for my most common queries (add (takes a table name and a hashref), update (ditto), and so on), but if I need to do something I don't have a function for I can always hand-build a query on the db handle.
• WWW::Mechanize saves a lot of time. If you do a lot of this, you might also want HTML::Tree as a companion. The look_down method is pretty keen.
• Archive::Zip is spectacularly useful, especially for dealing with OpenOffice documents. (Next question: will I eventually pick an XML module similar to HTML::Tree and use that in conjunction with Archive::Zip for parsing, modifying, and saving my documents? So far I'm still dealing with the XML by hand...)
• Data::Dumper is a really useful debugging aid. There's also Data::Dumper::Streamer
• Clone::PP is helpful. The most common things I clone are DateTime objects.
• Finally, I have a handful of my own include files that I wouldn't want to be without; I copy them to every system I use Perl on. These aren't modules per se, as I 'require' rather than 'use' them.