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 update
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 update
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 update
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.