I generally use wget myself. For basic HTTP debugging needs, I run ":%!wget -Sd http://www.example.com inside a new vim buffer. Then I can read the Varnish headers or whatever and figure out what's going on.
I ran into an issue where the SSL implementation was a bit dated, though, and didn't recognize how a GoDaddy cert implemented multiple hostnames -- but it turned out to follow the standard. wget was just lacking in its implementation and reporting an error when the cert was fine.