If possible, I would suggest using WWW::Mechanize, which subclasses LWP::UserAgent. This would afford you access to cookies and the other functions of LWP::UserAgent, only better and simpler; plus extensive examples (at least three of which use cookies) and many articles for reference. Nodes using WWW::Mechanize abound here on PM; Super Search will undoubtedly turn up much helpful information.

Update: Also, as Corion notes here, helpful modules such as HTTP::Recorder and WWW::Mechanize::Shell exist to help you automate the creation of your WWW::Mechanize scripts.

HTH,