Install GNOME Shell normally as you would (ignore the change DE at login screen bit).
I assume so far you logged on using the startx command. So once you installed GNOME 3, try logging on using gnome-shell (or try gnome-session if that doesn't work).
PS: You might also want to look at this question, maybe it will be helpful to you.
To be honest it would be better to back up for files and all config files in the Home folder and do a fresh install of Ubunutu.
If not try this:
Start by running this command:
Uninstall just ubuntu-gnome-desktop
sudo apt-get remove ubuntu-gnome-desktop
sudo apt-get remove gnome-shell
This will remove just the ubuntu-gnome-desktop package itself.
Uninstall ubuntu-gnome-desktop and it's dependencies
sudo apt-get remove --auto-remove ubuntu-gnome-desktop
This will remove the ubuntu-gnome-desktop package and any other dependant packages which are no longer needed.
Purging your config/data too
If you also want to delete your local/config files for ubuntu-gnome-desktop then this will work.
Caution! Purged config/data can not be restored by reinstalling the package.
sudo apt-get purge ubuntu-gnome-desktop
Or similarly, like this ubuntu-gnome-desktop
sudo apt-get purge --auto-remove ubuntu-gnome-desktop
Then run:
sudo apt-get autoremove
to remove extra packages and dependencies no longer needed on your system.
You might (or might not, depending on if gdm was removed already) need to use these commands to change your login screen back to Unity's default (which is lightDM) run
sudo dpkg-reconfigure gdm
Hit enter at the prompt and then select lightdm from the options. After that you can remove gdm with
sudo apt-get remove gdm
Finally run:
sudo apt-get install ubuntu-desktop
sudo apt-get install unity
INFO HERE and HERE
Best Answer
That's because you haven't installed it yet :)
Open a terminal with Ctrl+Alt+T and enter: