How can I split the standard Ubuntu terminal?
Not with the standard terminal. There are, however, a number of applications that you can install that will extend your terminal to allow for it, called multiplexers. Tmux and Screen are two of the most common.
Use terminator / termssh
apt-get install terminator
# for GNOME but works fine on KDE.
https://github.com/vahidhedayati/termssh
wrote this to automate server connections either through autodiscovery or text file 4 -8 windows per tab
Like Shauna suggests, GNU Screen among other tools will let you do this. It also comes with a host of other features which can be very useful (like keeping your programs running on a remote shell even after you have disconnected, so that you can re-connect to the session later).
If you decide to try any of these tools out, I suggest you look a little bit further than your initial needs requires you too, as the rewards will be immense.