I've created a new user by using "User Accounts" in "System Settings". How do I grant rights so that this new user can have the same level of access as root?
You don't want to do that. Use sudo, instead.
On recent versions of Ubuntu, the default group that is allowed to elevate using sudo is (appropriately) named "sudo". You can add the user by using
usermod (or your graphical interface, which sadly I don't know anything about, or I would try to give you instructions for that).
usermod -a -G sudo new_user
Have the new user log in again. They should be able to
sudo to other users after that.
root - there can be only one.
(but root can have multiple names).
To create an account with the same priviledges as root, just make the UID of the account equal to zero. Then they will be root, but with a different name.
Common examples of this are account names like
toor which are created (e.g. on some *bsd systems) with UID 0, as an alternative root login - perhaps with
/bin/csh rather than
/bin/sh as the shell or some other frivolous reason.
However, as several people have already said, you do not want to do this. You may think you do, but you really don't. Install and configure