gvim or vim?

What can I say, these two are pretty much the same things sharing the same core; GVIM is just vim with a GUI wrapper, with a little extra goodies. Couple things come to mind are -- Gvim has better color support, this is obvious, because vim is a console app; also, gvim works better with tmux for sure, which I suppose isn't something you'd care about as you begin the journey of learning vim.

Mostly importantly, Gvim, being a GUI application, has these nice menu and icon bar there so that when you get overwhelmed in the ocean of vim keyboard commands, there's this option to fallback to mouse clicking to get some tasks done. But as every vim experts on the planet would tell ya, the secret of archiving superb vim productivity when cutting code is to keep your hands off the mice.

just learn vim or what ever server you ssh in.

I just use nano. No point. Unless you want to develop in vim and stop using your mouse. More power to ya.