Fate/Zero is a fantastic series, great plot and insane characters, just amazing in general.
However, I'd recommend going through Fate/stay night first. As it's technically the original it's probably best to go that way in the first place. Most of Saber's character development pretty much rides on the notion that you know her deal in Stay Night upon watching Zero in the first place, so very little is explained about her. Not to mention that if you wanna go through the whole series, Stay Night's anime is very underwhelming in comparison to Fate/Zero, so jumping from Zero to Stay Night is just an awful idea.
Also, watching the Unlimited Blade Works movie is a good idea. However, you HAVE to watch the TV anime first, otherwise there's no possible way you'll enjoy it. It's so fast-paced and condensed that anyone with no knowledge of Fate would be lost.