I have the ubuntu installer on a flash drive and I am trying to figure out a way to install Ubuntu from that flash drive onto my 32gb flash drive. My end goal is to have the 32gb drive hold the entire OS and be the boot drive because the computer I am doing this on has no HDD. Can anyone point me in the right direction?
Sudodus's advice is correct, doing a full install to USB device is just about the same as doing an install to internal drive. I have tried most of the things on the page he references with success. Imaging a Fully installed OS to USB simplifies installation a lot but you need the space to download the image file.
A few hints if you are building the drive from scratch:
If you ever plan on using the 32GB drive for data, on a Windows machine, you might want to use "something else" when partitioning and make the first partition NTFS.
If you are worried about wearing out the drive with too many writes, you can make a "/home" partition with an ext2 non-journaling file system.
While in the partitioning window you can confirm the 32GB drive is selected for grub.
It might be worth doing an encrypted install if you will be carrying the drive around with you.