Currently you are able to watch "Fate/Grand Order - Absolute Demonic Front: Babylonia" streaming on Funimation Now, Crunchyroll, HiDive or for free with ads on Crunchyroll, VRV. It is also possible to buy "Fate/Grand Order - Absolute Demonic Front: Babylonia" as download on Amazon Video, Microsoft Store.
So if you want to watch the Fate anime series in order you need to follow this:
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If you want to follow the “proper” order, you should watch first Fate/Zero, then Fate/Stay Night UBW from ufotable (not the movie version) and finally the movie Fate/Stay Night: Heaven’s Feel. However, ufotable put a lot of references in FSN: UBW that will fly over your head if you haven’t watched Fate/Zero.
If you want to watch the Fate/ series in chronological order you'll need to watch them as follows:
I just finished watching this about 20 minutes ago and I must say I have been anticipating this for a while now and this is one of the anime appearing this year I was really, REALLY excited to see. I must say I was a little disappointed. For starters I am a huge fan of the Fate series.
The Fate/Series comprises of different iterations of the same story, parallel worlds with different timelines, same worlds at different times (centuries or millennia apart), spinoffs, and some are barely related by the series name and the character's looks.