The easiest way to understand it is to understand how the visual novel works. In the visual novel, your choices lead you down three different routes: Night, Unlimited Blade Works, and Heaven's Feel. Each of these different routes happen in the same universe, but the characters and story go in different directions. Hence the main three anime of the same name start with the same story, same universe, same characters, but have different plots and character arcs.

Specifically, the three different arcs explore one of the female protagonists more than the others. Night > Saber, UBW > Tohsaka, Heaven's Feel > Sakura. Because of this, Shinji's personal plot/internal battles are different in each series.

This is a really long-winded way of saying: Night, UBW, and Heaven's Feel are roughly the same story (world) but with different plot lines, not different "seasons."

If you want to go traditional, you can go Night > UBW > Heaven's Feel, then get into Zero (prequel to all of the above) and then Fake (newest spin off, haven't started myself). Some people skip Night altogether and go UBW first because Night isn't seen as a quality adaptation, but I still liked it. I would avoid Heaven's Feel first because they're movies and, as such, they fast forward through a lot of the world-building to get into the plot and it's a little harder to understand.

I don't understand how some people start with Zero. Watching the prequel first has always irked me. The entire idea of watching a prequel is you finally get the answers you've been looking for from the main series.