Gersen: Theoretically Fate Zero is a prequel of all three (FSN, UBW and Heaven's Fell) as all three are "alternate" stories taking place at the same time with the same characters but different plots. (If you want to keep it "vague" and spoiler-less you could say that : FSN focus on Saber and Shiro story, UBW focus on Archer and Rin story while HF focus on Illyasviel story)
But practically Fate Zero is closer in plot and "mood" to UBW than it is to the original Deen's FSN and IRC Ufotable has said that UBW was meant to be the direct sequel to Zero.
Honestly, you should probably see the first Fate/Stay night anime. It is not very good, and it includes some stuff from other scenarios that could be considered spoilers, but it is the prologue. Most other stuff will assume you already know the characters and such.
Unlimited Break Works has an "episode 0" that last 40 minutes, that (according to my friends' tales) follows the game's prologue really well. If so, you should start with that before the main anime.
After that... I can't say for certain. I'd go with Unlimited blade works, the second scenario, and then Heaven's feel (if already available when you actually watch it). Fate Zero is a precuel, but it was meant to be read (the novels) after having played the game, so it will include tons of spoilers that should normally be told during the main story. I'd advise you to leave that one for last.
If after that you want more, go with Carnival Phantasm, it's a really funny take on the whole lore. And you have the magical girl Ilya spin off, I've read the manga and found it surprisingly interesting.
Niggles: Out of curiosity how different are the three alternate stories?
Long/pretentious explanation:
The first scenario, Fate, is basically an introduction. It tells a story of how things would normally go as well as setting heroes and Shirous beautiful ideals. The second one, unlimited blade works, is a pure shonen action fest that challenges what a hero really is as well as Shirou's ideals. The third one, Heaven's feel, it's quite the drama and explains all the underlying questions you have been building about the holy grail war. The third one is where Shirou dies most, so I like it a lot.
Short explanation:
It changes which girl you bang.