This took way longer to write up than I thought it would, since there’s more of this stuff than I remembered. Ugh.
The whole Fate franchise originally came from the Fate/stay night visual novel (which never got an official English release, still a surprise to this day as it would be a license to print money), so you’ve got a few series based on that, and then spin-offs of varying levels of legitimacy. The quick primer is probably along these lines:
Fate/stay night (TV, 2006) - The original Fate anime adaptation by Studio Deen from well over a decade ago. This is kind of the black sheep of the core Fate anime pantheon since it tries to combine all three of the main routes from the VN (Fate, Unlimited Blade Works and Heaven’s Feel) and doesn’t really do a great job of it. Decent, but largely obsolete given its relative age and it’s jumbled attempt to Do Everything.
Fate/stay night: Unlimited Blade Works (movie, 2010) - Studio Deen’s attempt to make a movie focusing on the core of the Unlimited Blade Works route from the original VN, this thing is completely obsolete in the modern era and was never very good in the first place. Hard pass in today’s age.
Fate/Zero (TV, 2011) - Based on a prequel novel to the visual novel, Fate/Zero was handed off to a different studio (ufotable) who previously adapted another Type Moon property (Kara no Kyoukai) and did a phenomenal job with that. From Fate/Zero onward, ufotable did much of the rest of this list. Fate/Zero is strange in that it’s a prequel written well after the original series by a different author, so it is a passable introduction to the franchise but largely expects the viewer to have a basic understanding of the world, the Holy Grail War, and many of the characters right from the start. Newcomers won’t get all of the references and such, but I’d still probably say this is the best starting point if you have no idea what you’re getting yourself into as this series is pretty solid. You may, however, be better served by starting with…
Fate/stay night: Unlimited Blade Works (TV, 2014) – Created by ufotable as something of a “sequel” to their own Fate/Zero, this is an adaptation of the entire Unlimited Blade Works route from the original VN that also links itself more heavily into Fate/Zero. Misgivings about some of the pacing in the latter half of the series aside, this is leaps and bounds better than F/SN 2006 at actually telling the baseline story of the original VN. F/Z will ruin some of the major reveals in UBW, but the same can be said of the reverse, so you have to take the hit on either one or the other.
Fate/stay night: Heaven’s Feel (3 movies, 2017-2020) – Again by ufotable, Heaven’s Feel is the third route from the original VN and this is the first real attempt to adapt the whole thing to an animated medium. I haven’t seen any of them yet since I’m waiting for the third one to be widely available so I can get them all in succession, but the more die-hard Fate fans I know swear by this as quite excellent. Only to be consumed after having seen some of the above, though, and definitely not in isolation as it assumes substantial core knowledge.
– Everything below this line isn’t really “Mainline Fate” any more –
Fate/Apocrypha (TV, 2017) – By A-1 Pictures rather than ufotable, Apocrypha is probably best described as a totally separate thing about a renegade Holy Grail War between two factions, with each faction getting a full batch of servants. This has some level of crossover with other elements of the greater franchise, but is largely meant as a standalone story. I didn’t much care for it, but if you just want more servants and the like, then you could do worse.
Fate/Extra: Last Encore (TV, 2018) – By Studio Shaft, this is based on (rather than a strict adaptation of) the Fate/Extra PSP game, featuring a virtual Holy Grail War being waged for control of a supercomputer on the Moon. Yeah, it’s weird. The series is also just not very interesting or engaging (kind of like the original PSP game). Play the remake game whenever that comes out in the year 20XX.
Fate/Grand Order (TV/movie, 2016 - ???) – Brought to you by every studio on the planet looking for money, these are all based on the Fate/Grand Order mobile game, where the grimdark future of 2015 has left mankind on the brink of extinction and their only hope is to send people back in time with artificial servants to find real servants and fight back against the forces of evil that seek to end the world. Also very weird, and requires significant knowledge of the mobile game that I neither have nor am able to offer to you in order to fully appreciate it.
Lord El-Melloi II Case Files: Rail Zeppelin Grace Note (TV, 2019) – By TROYCA, this series adapts a manga about Waver Velvet (of Fate/Zero) ten years after the end of Fate/Zero, the details of which I will selectively omit here. This is actually a pretty solid mystery series set in the world of Fate and is easily one of the better things on this list, featuring characters pulled in from Zero, Stay Night, Apocrypha, other Fate VNs that will literally never be translated (Hollow Ataraxia), Grand Order and other original characters. You don’t need to know any of them to watch, but just bear in mind that the mysteries aren’t Sherlock-types where the viewer can reasonably be expected to solve them independently. It’s more about the experience.
Fate/kaleid liner Prisma Illya (TV/movie/OVA, 2013 - ???) – An extended series from Silver Link based on an alternate universe where the Holy Grail War never really happens, characters get shuffled around, and a bunch of characters get turned into magical girls who collect servant cards. Different components of this franchise will vary from being light-hearted and fanservice-laden to some of the darker material in the greater Fate franchise. This series relies fairly heavily on high-level knowledge of the Fate franchise to understand most of the references that are scattered throughout the whole thing, and requires the capacity to put up with some extremely heavy fanservice elements compared to anything else on this list.
Today’s Menu for the Emiya Family (OVA, 2018) – From ufotable, this is basically a light, slice-of-life cooking series featuring the core Fate cast absent all the antagonism and any logical justification for everyone getting along. Watch Shirou and other characters cook stuff and generally be endearing for 13 half-length episodes. That either appeals to you or it doesn’t.
Carnival Phantasm (OVA, 2011) – A celebration of all things Type Moon (including Fate as a huge portion of it) from studio Lerche, Carnival Phantasm is an irreverent parody series that basically throws people from every Type Moon property (Fate, Tsukihime, Melty Blood, Kara no Kyoukai, etc.) into situations together for no other reason than comedic effect. Mileage is determined entirely by how familiar you are with anything that shows up within. Also has one of the catchiest opening songs of all time.