Install Unetbootin
You can probably install stuff from the terminal, so do CTR + SHIFT + T and type (or copy + paste) :
sudo add-apt-repository ppa:gezakovacs/ppa
sudo apt-get update
sudo apt-get install unetbootin
This installs Unetbootin, which allows you to boot a usb drive.
Install Ubuntu To USB Drive
First, make sure your drive is formatted. (with no files on it)
Open Unetbootin, make sure the dot is on "Distribution", then click on "== Select Distribution ==".
A menu will come down. Find and select "Ubuntu" from there and the menu will close.
Click on "== Select Version ==" and select "16.04_Live" or "16.04_Live_x64" depending on your hardware.
Then make sure that at the bottom, the type is set to "USB Drive", and that the "Drive :" menu is the correct one you want to install Ubuntu on.
Click "OK" and wait for it to install live Ubuntu on your drive!
Once it is done, shut down your computer and proceed to the next step.
Booting Live Ubuntu
Usually in a computers firmware, there is an option to choose what device or partition you want to boot from. Find your computer model, and look for how to do it with your particular hardware.
Turn on you computer, and press the desired key right when your computer logo comes up (not the Ubuntu or Mint symbol) and select boot from USB. It will show you a menu on boot, select "Try Ubuntu Without Installing".
Installing "Os Uninstaller" and uninstalling Linux Mint
Open the Terminal and type the following commands :
sudo add-apt-repository ppa:yannubuntu/boot-repair
sudo apt-get update; sudo apt-get install -y os-uninstaller && os-uninstaller
Open Os Uninstaller, and make sure it is uninstalling linuxmint. Click apply, and Mint will be gone.
Installing Ubuntu 16.04
Now open Install Ubuntu 16.04, and go through the menus and make sure to select "Install alongside Ubuntu 14.04", when it comes.
Booting Your System Again
Once it is installed, reboot your system, (make sure to select your hard drive for boot), and you should be able to select "Ubuntu 14.04", "Ubuntu 16.04" and a few other options.