If you want to experience the Fate story in chronological order, start with Fate/Zero, then continue with the Fate/stay night routes. Fate/Zero is set during the Fourth Holy Grail War, ten years before the Fifth Holy Grail War covered in Fate/stay night.
Watch order
Fate/Zero (2011–2012) — The prequel and the chronological starting point.
Fate/stay night: Unlimited Blade Works (2014–2015) — The main route centered on Shirou and Rin.
Fate/stay night [Heaven’s Feel] I. presage flower (2017)
Fate/stay night [Heaven’s Feel] II. lost butterfly (2019)
Fate/stay night [Heaven’s Feel] III. spring song (2020) — The darker alternate route centered on Sakura.
Optional: Fate/stay night (2006) — The original adaptation of the Fate route; watch it if you want the classic version of the story, if you want to be it in order so watch it before Stay/Night