Hi there, and YES, if you want to see KI as the longtime fans did from Day One onward, I completely recommend playing through the story in order to see for yourself the in-game tension as the KI world braced itself for the event that gave us Shadow Lords.
Season 1 has 8 fighters, each with 3 different endings; one cannon, two not so much. They’re fun to play through, and if you’re not loving certain fighters, just breeze through it in easy mode for some insight into the interactions between characters and the setup for the future.
Just for giggles, here’s what triggers each ending:
A – Defeat your sixth opponent with an Ultra Combo
B – Defeat your sixth opponent with an Ultra Combo, but NOT the seventh!
C – Defeat your sixth opponent with an Ultra Combo on fighters six AND seven
Don’t feel pressured to unlock all three Season 1 endings for everyone if you aren’t into it, HOWEVER- if you have a favorite Season 1 fighter, you’re really missing out if you don’t do the following: Get all three endings for said fighter, then go back into story mode one more time on medium or harder, and make SURE that you finish off your sixth opponent with an Ultra Combo… You’ll thank me later. 
As far as Season 2’s story mode, I totally recommend that you play them all in order; even if it’s not one of your favorite fighters, because there’s a ton of context that leads up to Shadow Lords here. Great stuff!
After that, go nuts with Shadow Lords with your three favorites, but if you want more story, don’t add Rash, RAAM, or Arbiter to your trio of fighters. They’re just guests in KI’s playground, so they don’t get KI lore.
I sincerely hope that helps! Have fun, and FIGHT ON!!!
-Zenek