I would recommend using Wireshark, which has a "Follow TCP Stream" option that makes it very easy to see the full requests and responses for a particular TCP connection. If you would prefer to use the command line, you can try tcpflow, a tool dedicated to capturing and reconstructing the contents of TCP streams.
Other options would be using an HTTP debugging proxy, like Charles or Fiddler as EricLaw suggests. These have the advantage of having specific support for HTTP to make it easier to deal with various sorts of encodings, and other features like saving requests to replay them or editing requests.
You could also use a tool like Firebug (Firefox), Web Inspector (Safari, Chrome, and other WebKit-based browsers), or Opera Dragonfly, all of which provide some ability to view the request and response headers and bodies (though most of them don't allow you to see the exact byte stream, but instead how the browsers parsed the requests).
And finally, you can always construct requests by hand, using something like telnet, netcat, or socat to connect to port 80 and type the request in manually, or a tool like htty to help easily construct a request and inspect the response.