Set up a proxy setup pointed to a squid or whatnot that logs request. Use tcpdump on the router. If you have a rooted Android device you can use tcpdump there, or instrument the HTTP framework code to log requests, or...

But assuming you have control of your home network and aren't just using a Linksys thing out of the box, just logging into the router and watching traffic is by far the most straightforward.

Decompile the apk with one of a number of tools (apktool, baksmali). Least that would be what I would do.

Fiddler on your desktop machine - http://fiddler2.com/documentation/Configure-Fiddler/Tasks/Co...

I'm guessing Wire Shark.