Installing from the Software Center (11.10 and newer)
Open the software center search for "gnome shell" and install it, or just click this link: gnome-shell

After it installs, log out.
for 11.10
at the login screen click the little gear icon and select "GNOME" from the menu.

for 12.04
at the login screen click the icon show and select "GNOME" from the menu.


both
After that just put in your password and login:
The login manager will remember the last session you went into so you only need to select GNOME once, after that you can just login normally.
You can also use the fallback mode if you prefer an old-school experience:
And if you use auto logins you might need to tell lightdm to always use shell:
Command Line Instructions:
sudo apt-get install gnome-shell
Installing from PPA (11.04 only):
Using GNOME3 currently is only available for 11.04. This is a major release of GNOME so it likely will not be backported to older versions of Ubuntu.
Note: GNOME3 is provided as a tech preview for 11.04, it is not officially supported by the Ubuntu team. Here's another warning before you try this.
Using GNOME3 on 11.04 will upgrade major parts of the desktop infrastructure, if you are not familiar with how to do package management then we recommend you try this on a virtualized installation or a non-production system, reverting back to vanilla 11.04 is not trivial.
Specifically GNOME3 will migrate your application settings to gsettings, so if you want to try to rollback you'll get your old settings from before you upgraded.
As with all major desktop upgrades we recommend backing up your important data before getting started.
Graphical Instructions:
Launch the Ubuntu Software Center:
In the menu, select Edit, and choose software sources:

In the Software Sources window, click Add:

In the add window add the following text: ppa:gnome3-team/gnome3 and click Add Source:

Click close

The Software Center will now automatically update and add the GNOME 3 PPA. Navigate to the GNOME3 section:

Run the Update Manager and update your packages (note, the PPA will upgrade some packages)
In addition you need to select and install the following packages (click on the bag icon):
Then Log out and back in, selecting the GNOME Session in GDM (Todo screenshot)
Command line Instructions
sudo add-apt-repository ppa:gnome3-team/gnome3
sudo apt-get update
sudo apt-get dist-upgrade
sudo apt-get install gnome-session
sudo apt-get dist-upgrade
sudo apt-get upgrade
sudo apt-get install gnome-shell
Then log out and back in using the GNOME3 Session.
Where to report problems:
- Discussion about GNOME3 packaging (Mailing list link on the bottom left): https://launchpad.net/~gnome3-team
- Most problems and issues with this set of packages can go on this list, it should be the first place you report problems and issues.
- Bug reports that are NOT packaging related: https://bugzilla.gnome.org/
- If GNOME3 isn't working well it could be packaging related, so please don't just randomly report bugs to GNOME unless you are confident that it's an upstream GNOME problem.
- If you're not sure if you should report the bug upstream then ask someone either on IRC or on the mailing list above if you need help determining if something should be reported to GNOME.
Removing GNOME3 and going back to stock 11.04
Following these steps will remove the GNOME3 PPA and revert your packages:
sudo apt-get install ppa-purge
sudo ppa-purge ppa:gnome3-team/gnome3
While downgrading, if aptitude gives the following output:
Leave the following dependencies unresolved:
gnome-control-center-data recommends gnome-control-center (<< 2.91) [...]
Then type this when it asks you to enter yes/no: - gnome-control-center-data (Note the - in the beginning, its important). Then press enter, and then it shouldn't show any packages under Leave the following dependencies unresolved. Type y, and you should soon be downgraded normally to the Ubuntu 11.04 packages.
References: