Yesterday I set up Ubuntu on a 32GB Sandisk flash drive. I booted it up on my Windows 7 PC, and it works flawlessly. I haven't had any problems getting it to boot with USB priority on that machine, or accessing any data stored on the partition its using. Then I tried plugging the same USB into my Windows 8 laptop. I changed the BIOS so that USB has priority, but it still booted straight to Windows 8. I tried 3 different USB ports, and none of them worked. Does anyone know if any settings on Windows 8 might be affecting this, or what I may be doing wrong? I can post a picture of my BIOS if anybody cares to see.
You're not doing anything wrong, exactly - sometimes Ubuntu has trouble with the new UEFI technology, which your laptop has.
You can see instructions for installing and using Ubuntu on a UEFI PC here at the Community Ubuntu Documentation site. Basically, you want to burn your Live USB (preferably with Ubuntu Secure Remix, but regular Ubuntu should work as well), and before you try to boot from it, look in your BIOS for something that says "FastBoot/QuickBoot" or "Intel Smart Response Technology/SRT" and disable it. "Secure Boot", if you have it, might also have to be disabled. You should be able to install Ubuntu as normal from there, but if you reboot and get no Grub menus or you boot straight to Windows (or other boot problems), then you want to boot from your Live USB again and run Boot-Repair.
If you still have problems, then post back here.
Both 12.04.2 LTS and 12.10 should work with UEFI Secure Boot.
It depends on which Ubuntu version you are using. If you are using Ubuntu 12.04.2 or 12.10 and beyond, then it should work; otherwise, you will have to switch back to legacy-BIOS. If you search how to boot older Operating systems in UEFI Secure Boot, you will find the answer you are looking for, but it will require a lot of risky tinkering. If it wasn't for the whole UEFI thing, I would have moved permanently to Linux instead of being stuck with this Windows 8 thing that I don't even like.
Anyway, have fun and God bless.