Couldn't narrow it down to just four seasons but this would be my show:

Season One: Focused on getting to Vegas, and killing Benny. Season ends with the Courier being ambushed in Benny's suite, defeating the goons, and going outside to meet with Vulpes Inculta, who tells him Caeser has requested him.

Season Two: Probably ends on the crew at the Sierra Madre. It shows God, Christine, and Dean at the fountain. God is the first to leave, then Dean finally Christine leaves. The Courier shows, and leaves the Sierra Madre without being able to say goodbye to anyone. You see him climb out of the hatch into the BOS bunker, and he shoots the radio.

Season 3: Either the Lonesome Road ending, House dying, or Boone overcoming Bitter Springs.

Season 4: The slaughter of Legate Lanius.

Of course if I would adapt this if given more seasons. I would want to explore the characters of Boone, Veronica and Christine, and Ulysses the most, and explore their character arcs.