If you ask five people this question, you'll get six answers. There are conflicting opinions on this, and I'll try to outline the major options available in this answer. But first, a tl;dr

tl;dr

• Fate/stay night was originally a 2004 visual novel that had three routes - Fate, Unlimited Blade Works, and Heaven's Feel (in that order).
• The 2006 anime by DEEN mostly adapted the Fate route.
• The 2010 movie by DEEN adapted a highly condensed form of the the Unlimited Blade Works route.
• Fall 2014 and Spring 2015 saw another adaptation of Fate/stay night, this time by Ufotable, which adapts the Unlimited Blade Works route.
• At some point in the future, Ufotable will release a movie or series of movies that will adapt Heaven's Feel
• There is also a manga series that (I believe) adapts the Fate route, but I'm going to ignore it, because it's generally thought to not be that great.

I do not advise watching the DEEN F/sn anime (the TV series) it is poorly produced, both in terms of objective quality (art, animation, etc.) and in terms of faithful adaptation of the original work. If you really want to know what happens in the Fate route, I guess you could watch it, but honestly, lots of what happens in Fate is also referenced in UBW and Heaven's Feel, so you'll be able to piece it together after watching the Ufotable version of those two routes.

If you ask five people this question, you'll get six answers. There are conflicting opinions on this, and I'll try to outline the major options available in this answer. But first, a TL;DR

TL;DR

• Fate/stay night was originally a 2004 visual novel that had three routes - Fate, Unlimited Blade Works, and Heaven's Feel (in that order).
• The 2006 anime by DEEN mostly adapted the Fate route.
• The 2010 movie by DEEN adapted a highly condensed form of the Unlimited Blade Works route.
• Fall 2014 and Spring 2015 saw another adaptation of Fate/stay night, this time by Ufotable, which adapts the Unlimited Blade Works route.
• At some point in the future, Ufotable will release a movie or series of movies that will adapt Heaven's Feel
• There is also a manga series that (I believe) adapts the Fate route, but I'm going to ignore it, because it's generally thought to not be that great.

I do not advise watching the DEEN F/sn anime (the TV series); it is poorly produced, both in terms of objective quality (art, animation, etc.) and in terms of faithful adaptation of the original work. If you really want to know what happens in the Fate route, I guess you could watch it, but honestly, lots of what happens in Fate is also referenced in UBW and Heaven's Feel, so you'll be able to piece it together after watching the Ufotable version of those two routes.

#tl;dr

###Fate/stay night

###Fate/Zero

#You want to stick to watching anime

###But you already watched the DEEN F/sn anime

#You are open to anything

#What about all the other Fate/asdfghjkl stuff?

###Fate/hollow ataraxia

###Fate/kaleid liner Prisma Illya

###Other Fate/gibberish

###The rest of the Nasuverse

tl;dr

Fate/stay night

Fate/Zero

You want to stick to watching anime

But you already watched the DEEN F/sn anime

You are open to anything

What about all the other Fate/asdfghjkl stuff?

Fate/hollow ataraxia

Fate/kaleid liner Prisma Illya

Other Fate/gibberish

The rest of the Nasuverse

1. Watch Ufotable's version of Unlimited Blade Works (prologue, first half, second half [to air April 2015])
2. Wait for Ufotable's version of Heaven's Feel
3. Then, watch the Fate/Zero anime (first half, second half)

1. Watch Ufotable's version of Unlimited Blade Works (prologue, first half, second half)
2. Wait for Ufotable's version of Heaven's Feel
3. Then, watch the Fate/Zero anime (first half, second half)