One reason to learn the basics of plain vi (not vim) even if you are in the emacs camp -- if you find yourself needing to do something in a pinch on an unfamiliar unix machine, it's very rare to find one that doesn't have vi installed.
I love plain vi. I spent a few years using it exclusively, despite the availability of vim. It helps you realize how much you can do with plain vi and ex commands that you'd often rely on plugins for with vim.
There's always ed. (Which is surprisingly good.)
In college, I once spent a month using ed exclusively, to force myself to really learn to use it. I wouldn't really want to go back (except for the perverse challenge), but it did make me really appreciate the power of ed/sed/awk/perl and ideas like Plan9's sam/samterm.
If you can ssh into the remote box, you can edit files there via Tramp, even via sudo - C-x C-f /ssh:user@host|sudo:otheruser@host:/path/to/file, it'll prompt as needed for whatever credentials are required.
Still worth knowing the basics of vi, but Tramp has made it vanishingly rare at this point that I ever have to edit anything outside my main local Emacs session, which is nice.