Fate's a weird beast...when one of these threads pops up, you'll always get a bunch of people saying "It's a side-story" or "It's a stand-alone" implying that you can just watch it and it'll be fine...but that's almost always inaccurate; you're probably going to be completely lost, regardless of the "stand-alone" nature of the current story. All of these side-stories still build from basic knowledge about the universe, and none of them give you that knowledge very well if you don't have the foundational stories under your belt. Of course, if you're not someone who sweats the details, you may still enjoy it just fine...but that's not my personality. I hate going into a show and seeing things that I know are definitely references to something else (because they're presented in that "hey fans, look at this" way), but I don't know what the reference is to. That said, I think non-sequential storytelling is valid mechanism, and I think Fate leans into that very well. As such, the order I tell people who want to get into Fate is:
- original Fate/stay night series
- Heaven's Feel movies
- Unlimited Blade Works series (ignore the movie)
- Fate/Zero
Once you've watched those, you're good to watch anything else in whatever order and probably be ok...I do still think you should probably watch Lord El-Melloi before Fate/strange Fake, though.
Side note: I place Heaven's Feel before Unlimited Blade Works because I think it's more cinematic to watch in that order; lots of people reverse the two. The order I specify also has the benefit of giving you a little space between the partially repeated SN content at the beginning of UBW.
Side note 2: If you don't want to watch -all- of that, you could just watch UBW and Zero, then move on; they give you the info you need, and I think UBW is better than the original SN series...I just think the order I outlined above is better if you think you'll end up watching them all anyway, specifically because SN isn't as well done as UBW; it's a bit of a let-down to watch them in the opposite order. You must watch either UBW or SN before HF, though; those movies leave out a ton of stuff because you're expected to have already seen it before
Side note 3: While Fate/Zero does come first chronologically, I felt it was way better to watch it -after- the other 3. Again, it feels more cinematic, because it's providing the backstory to the things you likely had questions about from all the stuff you just watched; the non-sequential storytelling I referenced earlier