"To answer that question, it'll be helpful to understand Fate/stay night's structure. Fate/stay night the visual novel is split up into three parallel routes that tell one meta-narrative. The TV anime covers the first of these, Fate. The anime movie covers the second, Unlimited Blade Works.
Each of these routes plays a different role in the meta-narrative. Fate is largely concerned with Saber and works largely as setup for the world and the other two routes. Unlimited Blade Works and Heaven's Feel, the third route, are much more focused on Emiya Shirou, the main character of Fate/stay night. Most of the connections that Fate/Zero has with Fate/stay night are largely contained in the Heaven's Feel route, which has no anime adaptation.
Unless you're planning on reading the Fate/stay night visual novel, watching Fate/Zero first won't matter. Even if you are planning on reading it, I don't think it'll matter that much except for some surprises that won't be surprises anymore.
That being said, Fate/zero will spoil most of Fate/Stay Night for you.
Also, If you're worried about not understanding anything, Fate/Zero does a pretty good job of being self-contained (what with it being set ten years before Fate/Stay Night ) and all of the lore and mechanics are explained.
So in the end:
Fate/Zero is great at being it's own anime, but it would be best to either Watch the Anime (Fate/Stay Night) or (In my Viewpoint as something even better) reading the visual novel first in order to better enjoy it. [The Fate/Stay Night Anime Series is considered inferior to the original work and is really only a meager adaption.]