Ok, because I don't want to go through the entirety of dragon ball with vegeta(as that would take forever) I am instead going to give a long recap on what's happened so far.
Vegeta's Side:
So, Gohan found Vegeta as a kid(he's around 2) took care of him, and Vegeta hit his head. He's still a bit stoic, but he is really nice. He accidentally became an oozaru once, and killed Gohan without realizing it. A while later, Bulma finds him and they start looking for the dragon balls. They help the turtle, meet master Roshi, get the nimbus(which vegeta can ride), meet Chichi, Chichi joins their little hunt, but leaves a little before we meet emperor Pilaf.(Pilaf part is the same) Vegeta meets Krillin, trains with Roshi, and the tournament is the same. Red ribbon army happens, we meet Uppa, Vegeta helps Uppa(Baba part is the same), and 22 tournament is the same Krillin dies, Vegeta rages, Roshi and Chioutzu die, Vegeta rages more and beats up Piccolo. 3 YEARS LATER Vegeta doesn't look like a squirt anymore, and Bulma falls head over heels for him. 23 tournament is the same, except Chichi doesn't get together with Vegeta(nor did she ever want to(she gets together with Yamcha in the end. They have a kid who shows up later)). Vegeta fights Piccolo Jr. and spares him, and they live happily ever after-5 YEARS LATER. Vegeta and Bulma have a son named Gohan(who looks like canon Gohan) and Raditz appears, claiming to be Vegeta's brother. Raditz takes Gohan, Vegeta teams up with Piccolo and together they defeat Raditz, but Vegeta dies, there are two other saiyans who will arrive on earth in 1 year. Vegeta runs down snake way, trains with King Kai while the others train at Kami's Lookout, and Gohan trains with Piccolo.(btw Vegeta still has his tail)
Goku's/Kakarot's/The Other Saiyan's Side:
Bardock is the king, he has 1 kid(Kakarot(who was hit on the head as a kid(how? Why? We don't know) so he's pretty nice)) and Raditz is Vegeta's older brother. Vegeta(who is still an elite, just not royalty) was sent to earth by his parents(who, in this, are left unnamed) after they found out that Frieza was coming. Planet is still destroyed, and only Raditz, Kakarot and Nappa are left. They all live under Frieza, which leads to Kakarot being a bit traumatized(He's still kind of nice, just really quiet). They find out there's another saiyan out there, and Kakarot sends Raditz to get him(with specific instructions to NOT be hostile), They find out Raditz is dead(Kakarot knows he disobeyed orders) and prepare to go to earth(Nappa has a bit of a head start).
And so begins our story.