It would be best to watch at least one of the main shows first, just to understand the lore and setting, but there won't be any direct plot tie-ins or anything like that.
If you do want to just watch one show for the background, I'd recommend Fate/Stay Night Unlimited Blade Works if you like more shounen-y shows, or Fate/Zero if you prefer darker and grittier shows, with heavier themes. Either one works fine as an entry point.
I have seen most of Fate catalog available on Netflix like zero, UBW, apocrypha, FGO and last encore. Also seen the heaven's feel movies. It has been a few years, though. Good to know I can directly jump into this one.