If you're wanting to get into Fate you need to understand one key thing which is reused time and time again in the franchise, the 'Holy Grail War', and the best way to do that would be either watching Fate/stay night the Original Anime Adaption of the visual novel or the later remake adaption Unlimited Blade Works. 

This adaption follows Shirou Emiya as hes quickly dragged into the Holy Grail War. This adaption is not as strong and its production values are sigificantly worse then any other main fate adaption, if this puts you off you'd be best to just skip it and watch the Unlimited Blade Works adaption instead.

All later adaptions refer to Unlimited Blade Works instead of this one so your not missing out on much by choosing not to watch it.

Unlimited Blade Works is a 'remake' adaption that was much stronger and better animated by ufotable which adapts the same main story of Fate/stay night but instead following Rin Tohsaka (and less Shirou Emiya) during a seperate branch of the visual novel that has various story changes that occur from characters taking different choices. 

If you intend to continue watching Fate, Unlimited Blade Works is a must watch as its often referred to during the franchise.

Continuation / main series of Fate/stay night: Unlimited Blade Works Prologue.

Continuation of Fate/stay night: Unlimited Blade Works.

A Prequel Light Novel anime adaption set 10 years before the events of Fate/stay night: Unlimited Blade Works. Which follows Kiritsugu Emiya, (Shirou Emiya's adoptive father) as he challenges the 'first' Holy Grail War of Fuyuki City.

It doesn't matter which you watch first, but its best to watch the prequel after seeing Fate/stay night: Unlimited Blade Works adaption as you get a greater appreciation for the differences between the two Holy Grail Wars (and theres no Shirou Emiya).

Continuation of Fate/Zero.

Fate/stay night: Heaven's Feel is yet another adaption of the same Holy Grail War in Fuyuki City as Fate/stay night and Fate/stay night: Unlimited Blade Works adapting yet another branch of the original visual novel instead focusing on Sakura Matou with a more sigificant story change with a much darker tone. 

This adaption is best watched AFTER Fate/Zero and Fate/stay night: Unlimited Blade Works adaptions. 

Continuation of Fate/stay night: Heaven's Feel.

Continuation of Fate/stay night: Heaven's Feel.

Lord El-Melloi II's Case Files is a Light Novel spin off sequel anime adaption to the Holy Grail War events of Fuyuki City in Fate/stay night: Unlimited Blade Works which follows Lord El-Melloi II also known as Grail War Participant Waver Velvet as he investigates various supernatual mysteries. 

Other than needing to watch this after Fate/stay night: Unlimited Blade Works, it doesn't really matter and has no connection to anything else as its a far future set sequel. 

Continuation / main series of Lord El-Melloi II's Case Files {Rail Zeppelin} Grace note - A Grave Keeper, a Cat, and a Mage.

Fate/Kaleid Liner Prisma Illya is a parody spin off manga adaption to the original series which completely reworks the concept into a magical girls series. Still set in Fuyuki City it features very similar verisons of the main cast with the roles reshuffled and the main character being the young Illyasviel. 

Don't be put off by the fact its now a Magical Girls show, its equally as gritty and entertaining as the main series. The only anime watch needed before watching this is Fate/stay night: Unlimited Blade Works for an understanding of how Servants originally work as well as knowing the original verisons of the cast. 

Continuation of Fate/Kaleid Liner Prisma Illya.

Continuation of Fate/Kaleid Liner Prisma Illya 2wei!.

Continuation of Fate/kaleid Liner Prisma Illya 2wei Herz!.

Prequel to Fate/Kaleid Liner Prisma Illya.

Should not be watched until after Fate/Kaleid Liner Prisma Illya 3rei!!. This movie covers major character backstory which isn't explored until 3rei.

Continuation of Fate/Kaleid Liner Prisma Illya 3rei!!.

Fate/Grand Order: First Order is an alternate future spin off set after the events of Fate/stay night: Unlimited Blade Works. It adapts the story of the popular mobile game by the same title. 

This adaption is best watched AFTER Fate/Zero and Fate/stay night: Unlimited Blade Works. 

Fate/Grand Order - Divine Realm of the Round Table: Camelot Adapts the Camelot storyline of the Fate/Grand Order Mobile Game.

This should be watched after Fate/Grand Order: First Order.

Continuation of Fate/Grand Order - Divine Realm of the Round Table: Camelot.

Fate/Grand Order: Absolute Demonic Front - Babylonia Adapts the Babylonia storyline of the Fate/Grand Order Mobile Game.

This should be watched after Fate/Grand Order: First Order and ideally Fate/Grand Order - Divine Realm of the Round Table: Camelot. Theres little connection between Camelot and Babylonia so the order between the two doesn't actually matter.

Finale of the Fate/Grand Order mobile game anime adaptions. 

This should be watched after Fate/Grand Order: First Order, Fate/Grand Order - Divine Realm of the Round Table: Camelot and Fate/Grand Order: Absolute Demonic Front - Babylonia for maximum context available via anime.

I would highly recommend also watching other fate anime adaptations prior also as many characters make minor appearances during this finale. Adaptions with notable appearances include: Fate/stay night: Unlimited Blade Works.

Fate/Apocrypha is an alternate univerise Light Novel anime adaption of a completely different Holy Grail War with no real connections to any other adaptions. 

The only required watches before watching this is either the Fate/stay night or Fate/stay night: Unlimited Blade Works adaptions for that underlying understanding of how the Holy Grail War works. 

Fate/Extra Last Encore is yet another alternate univerise video game anime adaption of a completely different Holy Grail War with little connection to any other adaptions. 

The only required watches before watching this is either the Fate/stay night or Fate/stay night: Unlimited Blade Works adaptions for that underlying understanding of how the Holy Grail War works aswell as knowledge of the original versions of the reappearing characters.

Umu! 

Finale continuation of Fate/Extra Last Encore. 

A sequel Light Novel anime adaption set years after the events of Fate/stay night: Unlimited Blade Works. Following a "fake" Holy Grail War in the US.

The only required watches before watching this is either the Fate/stay night or Fate/stay night: Unlimited Blade Works adaptions for that underlying understanding of how the Holy Grail War works aswell as knowledge of the frequently mentioned original "Holy Grial War" AND Lord El-Melloi II's Case Files {Rail Zeppelin} Grace note which features some of the main characters that appear in the series.

I would highly recommend watching other fate anime adaptations prior for maximum context available via anime as many characters from other series make both main and minor appearances during this series. Adaptions with notable appearances include: Fate/Grand Order: First Order, Fate/Grand Order - Divine Realm of the Round Table: Camelot, Fate/Grand Order: Absolute Demonic Front - Babylonia, Fate/Grand Order Final Singularity: Grand Temple of Time - Solomon.

Continuation / main series of Fate/strange Fake: Whispers of Dawn.

Today's Menu for Emiya Family is a parody spin off manga adaption to the original series which completely reworks the concept into a slice of life cooking series. Still set in Fuyuki City it features very similar verisons of the main cast with the roles reshuffled into a family sharing meals together.

The only anime needed before watching this is Fate/stay night: Unlimited Blade Works so you know who the characters are and a general idea of their original relationships.

Fate/Prototype is an anime adaption of the original novel that later become the visual novel. It doesn't really mean anything and is more of a fan service adaption than anything else. 

The only required watches before watching this is either the Fate/stay night or Fate/stay night: Unlimited Blade Works adaptions so you can see what came of this original novel.

Carnival Phantasm is the first of various gag comedy adaptions of the fate and other franchises which basically pokes fun at itself and puts characters into silly and comedic circumstances. 

This show is not serious at all, but to appreicate it best you'd need to ideally see; Seperate franchise Lunar Legend Tsukihime and either the Fate/stay night or Fate/stay night: Unlimited Blade Works adaptions.

Fate/Kaleid Liner Prisma Illya version of Carnival Phantasm.

This show is not serious at all, but to appreicate it best you'd need to ideally watch Fate/Kaleid Liner Prisma Illya first.

Fate/Grand Order: First Order version of Carnival Phantasm.

This show is not serious at all, but to best appreicate it you'd need to ideally watch Fate/Grand Order: First Order, Fate/Grand Order - Divine Realm of the Round Table: Camelot and Fate/Grand Order: Absolute Demonic Front - Babylonia first.