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?
As an alternative, you may setup a Samba (SMB or CIFS) share on your RPi. See Samba for details. The advantages are that both Linux and Windows systems can read and write SMB/CIFS shares. There are also file managers on Android that also support these network file shares.
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:
You can also do this with plain old ssh using sshfs fuse
apt-get install sshfs sshfs user@rpi:/mnt/hdd /home/rpi/hdd
Run this on Ubuntu to make the rpi hdd available at /home/rpi/hdd.