If you've asked LWP to automatically follow redirects, it looks like the HTTP::Response request method will tell you the actual request that resulted in the page you ended up with, and you can use the previous method to walk backwards through all of the redirects that were done.

If it's not following redirects for you, you can use the is_redirect method to determine if the response was a redirect, then look at $r->header('Location') to see where the redirect was to.