Anime

Honestly, people make it out to be harder than it is. There are 3 mainline routes: Fate (the Deen anime), Unlimited Bladeworks and Heaven's Feel (both ufotable). These are the ones that actually matter.

Regarding the order:

1. Fate route. The Deen one. Bit harder to watch due to animation quality, but still a good time imo. Could arguably be skipped too, as it's not particularly faithful, but I'd still recommend it.
2. Unlimited Bladeworks (ufotable). Can't hurt to be at 1 either, but better to get the worst one out of the way.
3. Heaven's Feel (also ufotable). Absolutely has to be the last.

That's it.

Zero is generally recommended as well, because it's a really really good prequel. It can be watched before or after those 3, depending on who you ask (I watched it before and think I prefer it this way, but eh, opinions differ). 1st episode is very exposition heavy, but after that it's a great time.

All the other stuff is all based on the concept of these three routes, but aren't directly related. You can not watch it, and not miss anything. If you do want to watch the others, you do need the basis of the 3 routes, but they're all standalone from each other.

Most of the discussion is just opinion and centred around Zero and the Deen adaptation for a multitude of reasons, but its basis is not at all difficult or messy.

As someone who does (sorta) get the Fate series, here's a quick rundown for you or anyone else popping into this thread and who might be interested. Both on how it ended up like this and possible ways to get into the Fate franchise.

Quick history lesson to give some background on how it ended up like this: Kinoko Nasu's Fate/Stay Night released as a visual novel in 2004 and is set in the same world as other works they've previously released (e.g. Tsukihime, Kara no Kyoukai) none of these are mandatory to understand FSN but can flesh out the universe, also called Nasuverse.
FSN has three routes called Fate, Unlimited Blade Works and Heaven's Feel. They all split off very early on and branch in totally different ways. Now what made FSN special is that the three routes are meant to be experienced in that order as Unlimited Blade Works builds on what you learn in Fate. Heaven's Feel builds on both Fate and Unlimited Blade Works. So to fully understand the original work in its intended form, one would need to experience the three routes in that order. The original release and MOST but not all rereleases force you to experience the routes in that order.

Now why did I write all this? It's because FSN Fate never got a real adaptation. There is the Studio Deen adapation from 2006, which did mostly Fate and then mashed all three routes into one huge spoiler-mess towards the end that spoils the other two routes. This is the big thing stopping people from giving a clear answer to "Where do I even begin?"

In 2012 Ufotable released their adaptation of Fate/Zero, which is a prequel set 10 years before the events of FSN and co-written by Urobuchi (of Madoka fame) and Nasu. Sadly a lot of the plot points in FSN, especially in the later routes, are about finding out "what happened 10 years before now", so this prequel does have some spoilers for those routes and vice versa.
Later on Ufotable then also adapted *Unlimited Blade Works *as an episode 0 and two seasons. Even later they adapted Heaven's Feel as three movies.

With all that being said, the BEST way is to read the original VN,at least for the Fate route. The VN got a good remaster (first time officially in English too) this year on Steam.
However, I do understand that not everyone has the time or patience to read such a wordy VN. The two alternatives I can present are either substituting Fate with the Fate/Zero anime, which will leave you with some spoilers and questions, but it's not too bad. Or you rawdog the Unlimited Blade Works adaptation by ufotable as a starting point, which is less spoilery but probably a bit more confusing at the start. After Unlimited Blade Works, no matter where you started, it makes sense to watch Heaven's Feel and then, if you didn't already watch it, Fate/Zero. At that point you should have a broad enough understanding of what Fate is about to be able to enjoy any of the spinoffs without being too lost.

tldr to answer the question: "Where do I even begin?"
best: Read the original VN
otherwise:

1. Fate/Zero (ufotable) (prequel to substitute the missing Fate route adaptation, slight spoilers for HF)
2. FSN Unlimited Blade Works (ufotable)
3. FSN Heaven's Feel (ufotable, movie trillogy)

or equally good:

1. FSN Unlimited Blade Works (ufotable)
2. FSN Heaven's Feel (ufotable, movie trillogy)
3. Fate/Zero (ufotable) (prequel was written to be enjoyed after HF, originally)

After that you're really free to enjoy any Fate universe content. I could make some suggestions but I feel like this comment is getting too long already.

Addendum for people who are interested why Fate/Strange Fake (FsF) is so exciting: FsF is written by Ryohgo Narita of Baccano and Durarara fame. Originally he wrote one chapter as an April Fool's joke, but the people overseeing the Fate Franchise ended up offering him to write it as a light novel for real. I've read most of FsF that came out so far (there's one final volume scheduled to release in 2025). It is, in my opinion, one of the best Fate works written ever since Nasu's original VN. This is mostly thanks to Narita being really talented at writing a huge cast of characters and involve so many different viewpoints, which is very cool for a magical battle royale. Starting with FsF for interested newcomers isn't impossible and it is written with that in mind too, but it is very much something that gets more enjoyable the more the reader/viewer already knows, as there are many characters from other works that make cameos or references.

If anyone's curious about anything specific or if I was too rambly to be coherent in some places, feel free to reach out.

Small Edit: Ignore all the "timeline of Fate universe" memes. Most of them are inaccurate at best and often misleading. Think of it more like e.g. comic books used to be where MOST works are in similar universes but they don't need to be consumed sequentially to be enjoyed. For begginers, the only really important sequence is the Fate/Zero -> FSN -> FHA order, and FHA can be safely skipped and F/Z is a prequel that was originally meant to be read after FSN.

Fate/Zero is the best place to start in my opinion. And a fantastic show.

Edit: And if you wanted to watch it from "the beginning" you could try Fate/Stay Night UBW, but Fate/Zero is a prequel to that (and better).

I'd watch whichever came out first, then call it a day. The format/purpose of them all is the same as far as I know. Battle Royale.

I don't get the fate series at all...Where do I even begin?

Honestly, same. I haven't watched it, and, at this point, I'm not sure if I ever will.