What it says in the title
Zero is pretty great but I'd vote for watching Night. I haven't seen it, but it probably isn't as good as Zero and would end up as a major disappointment if you you went in expecting a Zero quality series but might pretty good if you go in expecting nothing.
Fate/stay night first (then Unlimited Blade Works), then Fate/Zero.
You're probably going to get lost story-wise if you jump into Fate/Zero first, so go with the original. Unlimited Blade Works is an alternate version of stay/night; you're free to watch that one (after stay/night of course), but I feel it's not really necessary overall.
Additionally IMO, the quality gradually just gets better if you watch it in that order, but that's just me. I thought Fate/Stay night was decent; UBW was better than the latter, and Fate/Zero is pretty awesome (so far).
You're probably going to get lost story-wise if you jump into Fate/Zero first, so go with the original.
I've read this a few times but I really don't think it's the case. I didn't see any Fate content before Zero and it's totally not confusing in the slightest. I love it but was kind of let down because I went in expecting this really deep and complex story because everyone said it wouldn't make sense, but it's a very straight forward Harry Potter wizard clans summon history's mightiest warriors and try to beat each other up in crazy shonen fights while sneaking in standard genre rhetoric about the power of friendship and believing in yourself. There's nothing particularly complex about it other than an eventual double cross that you could see coming from a mile away. It's a very straightforward series.
It's like Star Wars or Kingdom Hearts in how they placed their stories. The first story is set years behind the second story. If you don't want to spoil yourself, go watch Fate Stay Night first and then Fate/Zero.
I say Fate Zero is more exciting than Fate Stay Night, but Fate Stay Night has a very strong romantic theme later in the series. Watch for Implied Sex Scene. Of course, I was a naive teenager who didn't get suspicious at those certain scenes.
I watched Fate/Zero without having any prior knowledge about what happens in the Fate universe and can't say I had any problems undestanding what's happening. Sure later I watched FSN which pretty gave away what is going to happen later in FZ but don't really mind since FZ is great. Btw FZ>>>>>>>>>>>>>>>FSN.
There is seemingly no 'perfect' way to truly say goodbye. So I gathered a bunch of them!