This article is no longer relevant :)
Sure, you can use many ways to give Gnome Shell 3.7.5 (currently) a test drive, but in my personal opinion Ubuntu is the easiest, safest and fastest way to do it, accepting the drawbacks that come from the absence of systemd etc. And since we have decided to do some unstable software testing, why not do it on the unstable basis of the upcoming 13.04 release of the most popular Linux distribution in the world?
A ringtailed lemur eating a cookie containing the milk of a spherical cow
So, first thing you need to do is download the current Ubuntu 13.04 image and install it in your system. Note that the distribution is not anywhere near ready yet and you may encounter installation or other kind of problems, depending on your hardware and luck. You should install 13.04 on a virtual machine, or a small partition used for testing purposes and NOT for real work.
After you have successfully installed your new system, you will boot on a Unity environment :( You can install Gnome Shell right away from the distribution’s software center but the version will still be the 3.6 and will stay like that forever for this release (no GS 3.8 for 13.04).
The way repositories are added and used on Ubuntu, is the main reason why I believe this distribution is the easiest and fastest way to give GS 3.8 a try. You basically need to add two more repos, or three if you are adventurous!
The first repository is the Gnome 3 Team repo that contains all things missing from Ubuntu’s default repositories. You should add this one before the other two and run the usual update on your system. You can do this by typing the following commands on a terminal:
sudo add-apt-repository ppa:gnome3-team/gnome3 sudo apt-get upgrade
Then you need to add the Ricotz testing repository that contains cutting edge git versions for the shell, gtk, glib, clutter and many Gnome applications and utilities.
sudo add-apt-repository ppa:ricotz/testing sudo apt-get upgrade
If you want to get some more latest Gnome components and risk the stability of the system even further, then you can use the Ricotz staging repository that can be correctly used only if you added the previous two.
sudo add-apt-repository ppa:ricotz/staging sudo apt-get upgrade
After upgrading, you will have the latest available Gnome Shell version with many applications and utilities of the corresponding version. Note that some will still stay in 3.6.x version at least for now.
I have such a system installed for almost a month and tested it under various situations for a broad spectrum of tasks and I can tell you it feels much faster compared to Gnome 3.6 on Ubuntu 12.10. The animations seem to work better, the responsiveness is much better and in general it does feel a lot more stable than it was a month ago, but you should expect small issues here and there.
You cannot use any of the extensions, GTK or GS themes that you are using on Gnome 3.6 due to the usual incompatibility, but you could and should test your porting if you are a creator, to be one of the first to offer extending content for the 3.8 version. Whatever the case, you should go for it and tell us how it all went!