Saiyans work for Freeza, except Freeza uses their power to turn them into something similar to space cops. King Vegeta agrees, since Freeza's technology is beneficial to his people, but, deep down inside, he waits for an opportunity to overthrow Freeza. Eventually, he attacks Freeza with his elite Saiyan soldiers, as in the anime filler flashback. Freeza decides that this race is naturally evil and decides to destroy the planet, deeming it a necessary evil. Goku has been sent to Earth by that time, since Saiyans' practice of sending low class Saiyans to planets with weak beings exists in this timeline too, except they're not sent to destroy those planets, but rather to train there. Freeza decides to keep alive the four Saiyans that were not on Planet Vegeta when it was destroyed, mainly because he's impressed by Prince Vegeta's potential and, since he's still young, Freeza judges there is a chance he might have a change of heart if he receives proper upbringing from now on. Vegeta/Nappa/Raditz become Freeza's servants as in canon, except they're sent to planets to supervise, not to destroy. Vegeta secretly hates Freeza as in canon.
On Earth, Goku has not been brainwashed to kill humans, so Gohan has an even easier time raising him, though he's killed in the end all the same. Goku's life is the same as in the pre Z era until Raditz remembers him and arrives to recruit him. Since the Saiyans have been ordered by Freeza to behave, Raditz does not reveal his evil nature, so Goku's like, 'Chill out with us for a while, bro. I'll show you around.' Raditz accepts, without much enthusiasm nonetheless.
When, however, he's told about dragonballs, he reveals to Goku that his race have always hated Freeza and, since Goku is a Saiyan too, Raditz believes that he will share that hatred, so he proposes to Goku that they use the dragonballs to gain immortality and conquer the universe, thus restoring their race to their former glory. Goku refuses, Raditz kidnaps Gohan and the battle goes as in canon, with Goku dying and Raditz warning them about the other two Saiyans.
The rest of Saiyan Saga events are the same as in canon.
On Namek, Vegeta encounters Cui, who has been ordered to kill the prince, since Freeza has found out about the betrayal. Thanks to his Earth zenkai, Vegeta kills Cui.
Moori senses Freeza's good intentions and gives him the dragonball. Gohan and Krillin show and introduce themselves. Freeza says that he'll gladly let them use the dragonballs, when he has gathered them, to revive their friends and promises to deal with Vegeta when he finds him.
Still having their scouters, and the Nameks' help on top of that, Freeza and his henchmen find the last village before Vegeta can attack it. Then Dende (or some other Namek) takes them to Guru. Since there is no emergency, no one has their potential unlocked.
Porunga is summoned and everyone is revived.
Vegeta rides his pod and flees.
When Goku arrives on Namek, he's impressed by Freeza's power and challenges him to a friendly match. He has the upper hand, until Freeza uses his 2nd form's full power and reveals that he has two transformations left. Goku is excited that such a strong guy exists and invites Freeza to stay on Earth for a while. Freeza laughs and accepts, but also asks to use Earth dragonballs in order to find Vegeta, who is dangerous for the universe.
On Earth, Bulma has a crush on Zarbon, who reciprocates.
When Earth dragonballs are useable again, Freeza wishes that Vegeta is transported right before him. Seeing that Vegeta has regretted nothing, he kills him. Although Goku has his objections, he cannot stop Freeza; besides, everybody else agrees with Freeza on that one.
Nevertheless, Goku does not hold grudges and keeps training with Freeza. For simplicity sake, he reaches his post Ginyu base level.
Around the same time Freeza/Cold would attack the Earth in canon, Future Trunks appears. He's the son of Bulma and Zarbon, but, since his power is irrelevant, he never fought the Androids; he would just run and hide as everyone else. He gives the antidote to Goku and leaves.
Hearing about that future threat, Freeza decides to take his training more seriously. When Androids appear, SSJ Goku is at around Piccolo (pre Kami)/SSJ Future Trunks (second arrival) level, Gohan is his canon Freeza Saga self (post all zenkais) and humans/Piccolo are extremely irrelevant, since they were revived before they could spend any significant amount of time with Kai, if at all.
How the events go from here depends on whether you treat Super as canon.
1. Super is canon.
Freeza reaches his golden level and destroys all threats until Beerus wakes up. Goku becomes Whisses student and becomes SSJ Blue. Golden Freeza takes Vegeta's place in Zamasu Saga. Later, without Vegeta's help, Universe 11 most likely wins the TOP and Universe 7 remains erased.
2. Super is non canon.
Freeza calls his father for help. Over the following 3 years, Goku spars with Gohan while Freeza spars with Cold. Cold's power increases too, but he will soon be unable to catch up with the younger generation, unless Salagir warps this reality and transforms Cold into a giant with 1000 arms. Freeza unlocks a 5th form similar to Cooler's and, from now on, his power/gains are plot dependent, so that he remains Goku's arch rival. Freeza takes Vegeta's place during the first part of the Androids Saga (fights Eighteen). Cold takes Trunkses/Piccolo's place (fights Gero, perhaps goes with Gohan to Gero's lab later). Piccolo does not bother fusing with Kami, since he knows he's too weak either way. Cell absorbs people with no one disturbing him.
When Goku recovers, he brings up the ROSAT. Freeza nobly allows Goku/Gohan to go in first.
When the Androids arrive at Kame House, Goku/Gohan have already exited the ROSAT (since they stayed there only 21 hours in canon) and overpower them, until Cell arrives. Cell does not manage to talk Goku into letting him absorb the Androids, but he stubbornly keeps trying to absorb them and gets killed. The Androids retreat to live somewhere without bothering anyone anymore. Krillin gets no poon!
Freeza goes to the ROSAT too, because he doesn't want to be left that far behind, and unlocks a 6th form that makes him rival Goku's MSSJ, though Goku soon unlocks SSJ2 and leaves him in the dust again. Cold retires and becomes as relevant as Yamcha from now on.
7 years later, Dabura is easily disposed of.
