The Fate/Stay Night visual novel consists of three routes: Fate, Unlimited Blade Works, and Heaven's Feel. All three routes have the same beginning, but it isn't until after Shirou and Rin leave the Church after Kirei tells them how the Holy Grail War works that the plot becomes different between the three routes. While Shirou is still the main protagonist in all of them, each route of Fate/Stay Night has a different main heroine and a "main servant."
- Fate: Saber is both the main heroine and the main servant
- UBW: Rin is the the main heroine, while Archer is the main servant
- HF: Sakura is the main heroine, while Rider is the main servant
Because the plot is radically differs in all three routes, some servants and masters receive much more spotlight in one route, but receive little spotlight in another.
The other Fate/Stay Night TV series, the one made by Studio Deen in 2006, basically follows the entire structure of the Fate route with some story elements of UBW and HF mixed into it. Studio Deen's F/SN is not on Netflix. It's also a meh adaptation of the Fate route, and you're better off reading the VN if you want to experience the Fate route.
As for where to start:
Fate/Zero originally started as a light novel in 2007 that was released a few years after the Fate/Stay Night visual novel (which was released in 2004). However, ufotable adapted Fate/Zero first (in 2011-2012), then F/SN's UBW route (2014-2015).
While F/Z is the prequel to all three F/SN routes and thus spoils some plot twists from F/SN, there is only one plot twist in UBW that Fate/Zero spoils. When most fans talk about how F/Z spoils F/SN, it's the Heaven's Feel route that Fate/Zero spoils the most, because there are more plot twists that connect to Fate/Zero than the one plot twist in UBW. Ufotable will adapt the Heaven's Feel route next year.
That said, are you someone who doesn't like spoilers? Do you want to watch the Heaven's Feel movies coming out next year? Then it is probably best to watch ufotable's F/SN UBW adaptation first, then wait a year for the HF movies to come out, then watch Fate/Zero.
Of course, the earlier point about ufotable adapting F/Z first and then UBW should also be considered. Because ufotable adapted F/Z first, they had to make several changes here and there to satisfy those who watched F/Z first. That said, are you an anime-only viewer? Do you not care about spoilers? Then start with Fate/Zero.
So in summary:
- Fate/Stay Night generally refers to the visual novel, which consists of three different routes: Fate, Unlimited Blade Works, Heaven's Feel. The plot is far different in each route minus the beginning.
- Fate/Stay Night by Studio Deen is a meh adaptation of the Fate route and cannot be found on Netflix.
- If you are looking forward to watching the Heaven's Feel movies, plan on reading the visual novel, and/or you hate spoilers, Watch ufotable's TV adaptation of UBW first, then watch Fate/Zero after you see the HF movies. Just be warned that you'll have to wait at least a year for the HF movies to come out.
- If you are an anime-only viewer, a completionist who wants to watch everything and doesn't want to wait, and/or you don't care about spoilers, watch Fate/Zero first, then ufotable's Fate/Stay Night Unlimited Blade Works adaptation.
My personal bias would recommend watching Fate/Zero first, but that's because I'm one of those people who doesn't care about spoilers and will still enjoy something even if I'm spoiled about it in advance, but use the above information I provided and make your judgement there.