Fate/Zero spoils Fate/Stay Night, while Fate/Stay Night spoils Fate/Zero, but watch Stay Night first. Fate Zero assumes that you know what happens in the end, and shows the event leading to it, kinda like a classic Greek tragedy.
So either watch Fate/Stay Night UBW > Fate Zero
Fate/Stay Night 2006 > Fate/Stay Night UBW > Fate Zero
The 2006 version is slightly bad, but it can help introduce you into the world itself. An alternate route to the second option is Fate/Stay Night 2006 > Fate Zero > Fate Stay Night UBW, which is the order I watched it in.
After that, anything goes. And while you're at it, get a ticket for the Heaven's Feel movie if it's available in your state.