Q:

How can I add a new user as sudoer using the command line? 

After I add a user using adduser, I can't see it via System > Administration > Users and Groups unless I log out and then log in again.  Is that normal? 
Also, can I set a newly added user as a sudoer or do I have to change that only after adding it?  How can I do that via the shell? 
Finally, can I delete the original user that was created upon initial installation of Ubuntu, or is this user somehow 'special'? 

A:

Just add the user to the sudo group:
sudo adduser <username> sudo

The change will take effect the next time the user logs in. 
This works because /etc/sudoers is pre-configured to grant permissions to all members of this group (You should not have to make any changes to this):
# Allow members of group sudo to execute any command
%sudo   ALL=(ALL:ALL) ALL

As long as you have access to a user that is in the same groups as your "original" user, you can delete the old one. 

Realistically, there are also other groups your new user should be a member of.  If you set the Account type of a user to Administrator in Users Settings, it will be placed in at least all of these groups:
adm sudo lpadmin sambashare

Because your system configuration may vary, I suggest taking a look at the output of groups <username> to see what groups are normally in use. 

A:

I did
sudo usermod -a -G sudo <username>

as recommended here. 

A:

Open the sudoers file: sudo visudo will open the /etc/sudoers file in the editor defined in $EDITOR (probably GNU nano - set the variable if it's not what you want, eg export EDITOR="nano" and try sudo visudo again). 
Add the below line to the end of the file. 
username ALL=(ALL) ALL   # Change the user name before you issue the commands

Then perform WriteOut with Ctrl + O. The editor will ask you for the file name to write into.  The default will be a temporary file that's used by visudo to check for syntax errors before saving to the actual sudoers file.  Press Enter to accept it.  Quit the nano editor with Ctrl + X.
Done!