When you run the Ubuntu installer you have an option to choose the drive you are installing it to. The option will include all the drives that are available at the time you run the installer. If you have USB drives attached they will be included in the choice.
When performing the install, you'll also have the option of choosing the boot drive. If you choose the USB drive as the boot drive during the install, it would be just a matter of ensuring that drive is attached when you boot the computer, and use the computer's BIOS to boot to that drive.
The installation on the USB drive will be no different than and installation on your internal hard drive.
The drive choice for installation will come on the next screen after the one that has the option to
Erase disk and install Ubuntu. When you click continue, it will give you a second
install screen also titled
Erase disk and install Ubuntu. This second screen is the one that has will have the
Install Now button.
By the way, even clicking
Install Now, you'll still get one more chance to
Go Back. On that last change it will give you an information screen of which device (drive) will be changed and how it will be changed.
UEFI vs Legacy BIOS mode
You have to set your Computer's BIOS mode to match the installation type of UEFI or BIOS for the OS you are booting to.
To make your USB installation more universal compatible as you indicated in your question, you'll best choose Legacy mode for your installation. When I install to a USB drive, I only have the option (from the computers I have used so far) to install in Legacy.