You want to read about WWW::Mechanize, the descendant of LWP that will help you complete your task. It's surprisingly easy. You'll start with three lines (from the synopsis) and have a simple 20 line program that downloads everything before you know it.

Update: Here's an example of pulling and writing a file directly to a file. I consider that task to be somewhat un-obvious (or at least, not obviously documented):

use strict;
use WWW::Mechanize;

my $mech = WWW::Mechanize->new;
   $mech->get("http://kernel.org");
   $mech->get( $mech->find_link(text=>"F")->url, ":content_file"=>"cur
+rent-kernel.tar.bz2" );
[download]