It would be best to watch at least one of the main shows first, just to understand the lore and setting, but there won't be any direct plot tie-ins or anything like that.

If you do want to just watch one show for the background, I'd recommend Fate/Stay Night Unlimited Blade Works if you like more shounen-y shows, or Fate/Zero if you prefer darker and grittier shows, with heavier themes. Either one works fine as an entry point.

It's a franchise that all stems from the Fate/Stay Night Visual novel. The visual novel had three different routes, which have their own names as the subtitles (eg Fate/Stay Night Unlimited Blade Works is an adaptation of the UBW route of the VN). There's also a darker and more serious prequel called Fate/Zero. Beyond that there are an absolute ton of spinoffs like this set in alternate timelines. Think of it like marvel comics, it's basically the same deal, and the same "asking for watch order is mostly pointless" rule applies.

To start out, you want to watch Fate/Stay Night UBW (on the shounen side) and Fate/Zero (much more on the darker seinen side) in any order. Then you wanna watch the Fate/Stay Night Heaven's Feel movie trilogy. Or if you're really keen then I still think the 100hr Fate/Stay Night VN is the best entry point, if you have the patience or are into VNs in general.

After that, watch whatever you want. Each unique spinoff series is a separate Fate/whatever, and can be watched without worrying about watching Fate/anything else. Starting with the original stuff is only really good because it establishes lore that the spinoffs will assume you know, but even they can be skipped if you really wanna get into a specific spinoff show.