Gigguk honestly ruined fate for alot of newbies because of how he made it sound so much more complicated than it is, I will give you the order I watched it in (I believe this to be the best order personally) Keep in mind this is MY order, I do not take the VN's into account as I myself have not even played them. If you want to consume the VN's to then ignore this reply.
Keep in mind alot of fate stuff is "not canon" in the sense that their is PLENTY of side stuff to consume, for example fate/stay night compared to fate/stay night UBW compared to fate/stay night heavens feel. Depending on who you ask will depend on what you are told but the general consensus is that fate/stay night is optional if you only care about the canon "true" stuff. However I will list my watch order ignoring this fact.
The order I consumed fate and what I believe to be the ideal order for a NON VN consumer:
Studio Deen - Fate/stay night (optional)
Ufotable - Fate/Zero (tied to stay/night as a prequel set 10 years prior)
Ufotable - Fate/stay night Unlimited Blade Works
Ufotable - Fate/stay night Heavens Feel (Movie Trilogy)
A-1 Pictures - Fate/Apocrypha (You can watch this at any point, it is a good entry point into the fate series in my opinion as it stands on it's own two legs also I believe it to be one of the best Fate animations)
SHAFT - Fate/Extra Last Encore (I watched this and still have no clue where it is placed because I never researched it, pretty entertaining though and can be watched on it's own seperate from stay/night)
Lay-duce - Fate/Grand Order First Order (First animation of the Fate/Grand Order series, set far in the future of everything else with time travel shit, so I would watch GO stuff AFTER all the stay night stuff personally. This animation is also movie length and is 1 episode)
Clover Works - Fate/Grand Order: Absolute Demonic Front - Babylonia
Signal M.D - Fate/Grand Order: Divine Realm of the Round Table - Camelot Wandering; Agateram (Movie)
Production I.G - Fate/Grand Order: Divine Realm of the Round Table - Camelot Paladin; Agateram (Movie)
Clover Works - Fate/Grand Order: Final Singularity - Grand Temple of Time: Solomon (Movie)
The stuff listed below can be watched at any point really in my opinion:
SILVER LINK - Fate/Kaleid Liner Prisma Illya (all seasons - look up the order, I cannot be fuckin bothered to type out the long ass names)
A-1 Pictures - Fate/strange Fake -Whispers of Dawn- (Pilot for the upcoming fate/strange fake anime, very good and I highly rec)
Then you have the spinoffs like fate carnival and the emiya cooking show and random bits of fate stuff like fate/prototype. You will need to look into those if you want to watch them, I cannot justify the effort it would take to gather them all up for you.
This is not an exhaustive list of all fate productions however this is the stuff I deem the most relevant if you want to get into fate/ via the anime side.
Obviously everybody has their own idea on what the best watch order is, same with shit like Star Wars or Magical Index, all I can do is answer the question to the best of my ability. I hope this list helps.
Best of luck and welcome to the fate/fandom, probably one of the better ones especially considering the shit show that is One Piece and MHA.
(Sorry for inactivity, Have been drafted for my "required military service" so have no time to do anything, on leave this weekend to see family but will not be around often, shitty country making me do this shit, should have pulled the disabled card)