Firstly, welcome! Have you read the PerlMonks FAQ? Secondly, do you have to use LWP for this? The WWW::Mechanize module has a method called $mech->links which lists all the links on the page your looking at. The module documentation has good examples of how to go about using it.
If you have to use LWP have a look at the HTML::LinkExtractor module. "HTML::LinkExtractor - Extract links from an HTML document"