Start with Fate/zero, then look at Stay night or Unlimited Bladeworks before moving over to aprocrypha.There's been so many Fate/ animes's in Netflix. Any of the good? Fate/Stay night, maybe?
Story-line wise:
1. Fate/zero
2. Fate/stay-nigth
2.1 Fate/unlimited bladeworks (told from another point of view, small changes to the story but still fairly much the same)
Later on in a alternate universe:
3. Apocrypha
Not watched Prisma illya myself, as I thought it was more loli/hentai stuff from the few clips i saw on anime-reviews, but I think that again is in another alternate universe.
Heavens feel will come out eventually, which will be a alternate 3 where the ending of stay-night or unlimited bladeworks is changed drastically to tell another story.
At least this is from memory, could be wrong.
If you can stomach the depressing setting in Fate/Zero, you'll be fine with the rest as nothing of the rest is remotely as dark.