There are three common ways to install a modern linux to a usb.
One is using a hybrid iso and the command dd to copy the image to the drive. You don't want that in this case.
Second is the many how to's and tools at www.pendrivelinux.com
. I don't think that is what you want either.
For you on a 32G, I'd suggest that you power down your system and remove the power to the hard drives. Install your usb and power up to bios. Be sure that usb shows up in bios as a hard drive choice and first in order or hard drives. Be sure that cd/dvd is selected as boot device over hard drive device. Then boot to Fedora and install it as normal to the usb. It will not be as if it were the only drive and should work normally when booted directly on any supported computer.
My way is to use a virtual machine. Create a blank vm without a hard drive. Boot to iso image and attach usb then install. You can't reboot on a vm yet easily to a usb to you have to use real system.