How to install a Gnome desktop on Ubuntu?

1. Install the Gnome desktop environment on your instance.

sudo apt update && sudo apt install gnome-shell -y
Please note:

Do not install the ubuntu-gnome-desktop package. Doing so will install other unnecessary packages that will permanently break your instance.

2. Create a security group rule for TCP for VNC ports 5900-5910 and add it to your instance.  See the security group guide here.

3. Install a VNC server on your instance.

sudo apt install tigervnc-standalone-server -y
Please note:

See this wiki for advanced configurations. 

Download and install a VNC client on your local system. See RealVNC Viewer.

4. Start a VNC server session.

vncserver :1
  •  Killing your VNC session is done with
vncserver -kill :1

5. Connect to your desktop using the information provided by the VNC server command. 

For example, you should see a port number that you can connect to.

vncext:      Listening for VNC connections on local interface(s), port 5901