I have a Pi 3 running Raspbian with an external HD at /mnt. It is connected via ethernet. I would like to regularly be able to save files to that directory from another computer on my Wifi network running Ubuntu 16.04. What would be the best way to set up for regular file transfers?
Check out NFS. You would use NFS server on your Pi 3 to set up a share on your external HD. You would then use NFS client on your Ubuntu to do a mount of the remote disk share. Sorry if my terminology is not 100% and I can't dictate commands off the top of my head, but conceptually this would work.
Here are links to a couple guides that should help you find the relevant info quickly: