Vegeta walks into the story in Dragon Ball Z and never leaves it again: he is in every series that follows, from GT to Super and Daima. Vegeta's profile covers him by theme; here we follow him saga by saga, watching the conqueror who came to wipe out humanity turn, one relapse at a time, into a family man who fights to protect it.
Dragon Ball Z
Saiyan Saga (episodes 1-35)
The prince already shows up in episode 5, but the Saiyan Saga only brings him to Earth after a year in space, with Nappa, to wish for immortality on the Dragon Balls. He lets his partner do the fighting and, once Goku has crippled him, finishes Nappa off himself in episode 29. The duel with Goku is the heart of the arc: pushed back by the Kaioken, Vegeta makes an artificial moon and becomes a Great Ape in episode 32. It takes Yajirobe, who cuts off his tail, Krillin and a Great Ape Gohan to bring him down. He escapes in his pod because Goku asks Krillin to spare him.
Frieza Saga (episodes 36-107)
Healed, he heads for Namek to beat his master to the wish. The Frieza Saga casts him as a ruthless third party: he wipes out a Namekian village, kills Dodoria, who reveals the truth about Planet Vegeta's destruction in episode 49, and gets his revenge on Zarbon in episode 57, stronger for having nearly died. He grudgingly sides with Krillin and Gohan against the Ginyu Force, then takes on Frieza's final form and is torn apart. In episode 86 he dies in front of Goku, after telling him that Frieza exterminated the Saiyans and begging him to avenge them. The Dragon Balls bring him back on Earth; in the anime he then searches space for Goku, without success.
Androids Saga (episodes 118-139)
While waiting for the androids he trains to exhaustion under three hundred times Earth's gravity, until the capsule blows up around him. The Androids Saga gives him the moment he has always chased: in episode 129 he arrives as a Super Saiyan, saves Goku from the heart virus and destroys Android 19. It does not last. Android 18 takes him on alone and humiliates him, and Vegeta walks away from the group.
Cell Saga (episodes 140-194)
He trains with Trunks in the Hyperbolic Time Chamber, and the Cell Saga sees him come out with a new level of power: in episode 156 he calls himself "Super Vegeta" and dominates Cell. Then pride takes over. He wants an opponent at full strength and lets Cell absorb Android 18. Against Perfect Cell even his brand-new Final Flash only blows off an arm in episode 162. At the Cell Games, when Cell kills Trunks, he charges him in a blind rage in episode 189, and Gohan is hurt saving him. At the very end his blast distracts Cell and gives Gohan the opening he needs.
Majin Buu Saga (episodes 200-291)
Seven years later, the Majin Buu Saga holds both his worst mistake and his redemption. Dabura senses the evil in his heart and Babidi gets into his mind: episode 228 brings Majin Vegeta, who kills spectators at the tournament to force Goku into a duel and then admits he let himself be controlled on purpose. Their clash is what wakes Buu. Vegeta knocks Goku out, faces the monster alone and, in episode 237, blows himself up to protect his family, without managing to destroy it. King Yemma gives him back his body: with the Potara earrings he and Goku become Vegito, and later, on the Supreme Kai's world, he admits Goku is number one in episode 280 and comes up with the final Spirit Bomb, powered by the Earthlings he has revived with the Namekian Dragon Balls. The dragon brings him back to life too, a sign he no longer counts as evil.
Dragon Ball GT
Baby Saga (episodes 17-40)
At the start of the series he is the one who sends Trunks into space in Gohan's place. In the Baby Saga the Tuffle parasite, having already taken over Gohan and Goten, seizes his body in episode 27 and turns it against Goku. Vegeta is only freed when the beaten Baby abandons his body in episode 39.
Super 17 and Shadow Dragon Sagas (episodes 41-64)
In the Super 17 Saga he kills Nappa again after his escape from Hell, but the new android knocks him out and nearly kills him. His comeback arrives in the Shadow Dragon Saga: Bulma builds a Blutz wave generator and in episode 59 hits him with it. Vegeta turns into a Great Ape, then a Golden Great Ape, and finally a Super Saiyan 4. With Goku he forms Gogeta against the One-Star Dragon, and in the finale his rival leaves him the job of protecting Earth.
Dragon Ball Super
Battle of Gods Saga (episodes 1-18)
The Battle of Gods Saga shows him terrified of Beerus, the god he once watched humiliate his father as a child. At Bulma's party he does everything to keep him calm, but when the god strikes his wife he snaps: in episode 7 he goes Super Saiyan 2 and attacks, to no effect. He takes part in the Super Saiyan God ritual and, in episode 16, wins Whis over as a teacher with Bulma's instant noodles.
Resurrection 'F' Saga (episodes 19-27)
In the Resurrection 'F' Saga he comes back to Earth with Super Saiyan Blue. In episode 27 he overpowers Golden Frieza, but the tyrant blows up the planet and Vegeta dies with it: Whis rewinds time and Goku settles the score.
Universe 6 Saga (episodes 28-46)
In the Universe 6 Saga he beats Frost and Magetta; in episode 37 he goads the young Saiyan Cabba into reaching Super Saiyan, then defeats him. In episode 38 he falls to Hit's Time Skip.
Future Trunks Saga (episodes 47-76)
In the Future Trunks Saga he trains his son and follows him to his timeline. In episode 56 he faces Goku Black, who turns Super Saiyan Rosé and badly wounds him; in episode 66 he accepts the Potara again and becomes Vegito, but the fusion wears off before the fight is over.
Universe Survival Saga (episodes 77-131)
He refuses the Tournament of Power while Bulma is pregnant: in episode 83 Whis speeds up Bulla's birth. In the Universe Survival Saga he breaks his limits with an evolved Blue form and knocks out Toppo, now wielding a God of Destruction's power, in episode 126. Jiren throws him out of the arena in episode 128, but before he falls he hands Goku the last of his energy. Later, in the Super manga, Beerus teaches him destruction and Vegeta unlocks Ultra Ego.
Dragon Ball Daima
The journey through the Demon Realm (episodes 1-20)
Shrunk by Gomah's wish, he works with Bulma on Shin's ship and then reaches the Demon Realm with her and Piccolo, as our Daima episode guide explains. In episode 12 he shows Super Saiyan 3 for the first time and wins a Dragon Ball by answering a riddle. Back to adult size, in episode 19 he takes on Gomah as a Super Saiyan 3 and ends the assault with a Final Flash, until an exhausted Vegeta is reined in by Bulma.
Movies
In the Super films he is always on the front line: in Dragon Ball Super: Broly he learns the Fusion Dance and becomes Gogeta with Goku; in Super Hero he stays on Beerus's planet training with Goku and Broly.
Our take
Seen saga by saga, Vegeta's story is made of defeats more than victories: Android 18, Perfect Cell, Hit, Jiren. Yet every loss moves him a step further. On Namek he dies asking for revenge for his people, in the Cell Saga he loses his head over a son, against Buu he sacrifices himself for a family. The writing works because the change never erases his pride: it redirects it.
Super adds the shade that was missing, the comic Vegeta who cooks for Beerus, without taking away moments of glory like Toppo's fall.
For his family, read Trunks saga by saga and Bulma saga by saga; for his eternal rival there is Goku's profile.