sudo vim /etc/apt/sources.list

change every occurence of "dapper" to "edgy" and save the file. You can use you own favorite editor of course, but I assume you are doing this from the command line because you are more comfortable there; hence, I show vim. Now you need to do a package database update with the following command

sudo apt-get update

As always with this command, if you see some failures, try to run the exact same command again. In my experience, with a slow connection or during a busy time, the repository requests will sometimes time out so running the command again will get you past it. Now, don't just run the thing over and over again until it works, because you might actually have a problem so it might not ever work without some debugging. Next we perform the first step of the upgrade.

sudo apt-get dist-upgrade

wait... wait... wait... and now do it again to pick up the "upstart" stuff.

sudo apt-get dist-upgrade

Now you'll need to reboot because you have a new kernel. And then you're done.