I am not writing this from the perspective of an expert but rather from someone who has spent the last couple of days doing the same things over and over again expecting different results. I cannot express how frustrating this whole installation process has been.

After booting in live mode, the installer for Ubuntu would keep on failing and launching the UEFI (modern BIOS) screen.

The final solution was simple. And for the sad human being who is searching this following phrase over and over again, you will hopefully find the answer below.

Installing Ubuntu on a Surface Pro 3 fails during installation.

Problem 1. Make sure that you set the resolution to 1920x1080 and set scaling to 100%

As dumb as it sounds, the installer apparently crashes as soon as it tries to load the image assets in the default high scaling high resolution setup that the Surface Pro 3 launches in

Problem 2. First install fails on the 'grub-efi-amd64-signed' package failed to install error message

Something apparently happens while the install is going on.

Follow the instructions that are on the link below to delete the grub file.

https://ubuntuforums.org/showthread.php?t=2112273

Once done, the boot-repair part of the instructions might still say there is an error. Reboot anyway and see if it fixed your issue.

It fixed mine. If your journey is any further, then sadly I can't help you.