The Fate series can be viewed, despite there being so much of it, as completely seperate from each other and doesn't really require you to watch one show to understand the other. I'll give a quick breakdown of all the various animes and where they fall in relation to each other.
The Fate/stay night series is based on a Japanese visual novel, which has 3 seperate routes depending on the heroine you chose. They had a 2006 TV anime of the same name dealing with the first route, a 2010 movie that was turned into a TV show in 2015 dealing with the second and an as of yet unfinished movie trilogy of the third. The cast is pretty much the same, but the shows are totally different. It is best to start here if you are totally new to Fate.
Fate/Zero is a prequel to stay night and while it is arguably the "best" show overall, watching this first will ruin a lot of the surprises and setups in the Fate/stay night shows. The Case Files of Lord El-Melloi II:Rail Zeppelin take place between Zero and Stay night, and is essentially a detective series featuring a Zero character and his interactions with background characters in the series. Fate/kaleid liner Prisma Illya is the magical girl spinoff show that takes characters from both FSN and Zero and uses them as the main cast.
Fate/Extra: Last Encore is a part of the alternate universe Fate/Extra series, which is almost entirely based on a series of videogames. Last Encore is a pretty different retelling of the events of the first game and is....seen as pretty bad by fans due to how off script the show goes. If you don't have any experience with the games your mileage may vary.
Fate/Apocrypha is another alternate universe and has no connection to any other series and can be enjoyed with no knowledge of any other show.
Fate/Grand Order is a japanese gacha mobile game in its own seperate universe from the other Fate shows, but characters from other shows will make cameos or appearances. First Order is a TV movie adaptation of the prologue of the game, and the currently airing Absolute Demonic Front: Babylonia is based on the 7th chapter of the game. A 2 part movie will be released in Japan sometime in 2020 dealing with the 6th chapter of the game.
Lastly, Today's Menu for the Emiya Family and Carnival Phantasm are shows that utilize the cast from FSN and are more about cooking in the case of Emiya Family and about in jokes and comedy in the case of Carnival Phantasm.
