A lot of things would stay the same, but altered for the sake of continuity.
I would actually employ Goku as an over-powered deus ex machina in the first "season". At first he has issues adjusting to his new (old?) body but quickly gets used to it during his adventures in space.
Baby Saga
The Black Star Dragon Balls would come from another dimension, same as the version of Pilaf we see make the wish. The Black Star Dragon Balls were spread to different dimensions to prevent someone from using them. The last Dragon Ball was located in the "main" universe where the story takes place. Pilaf still makes the kid wish and the Earth is still in danger of exploding. Goten actually joins Pan, Trunks, and Goku on the trip across the universe to find the black star dragon balls. Baby (bebi, bebe, etc. I'll use the funimation name)arrives on Earth and only possesses Vegeta and Gohan (who still has his elder-kai power-up). While the others are gone, Baby Vegeta and Gohan conquer the Earth. Uub tries to fight back but he's taken down fairly quickly. When they return from space Goku beats Gohan easily (illustrating how strong he's gotten post "DBZ"). Baby Vegeta absorbs Gohan's energy after Goku beats him. Goku loses, Supreme Kai intervenes before Baby Vegeta finishes him off. Goku discovers Super Saiyan 4 while on the Supreme Kai's planet (he gets his adult body back, laments what he calls a training opportunity loss). Fully transformed upon returning he makes quick work of Baby Vegeta. Baby purges himself from Vegeta's body and attempts to escape, Goku disintergrates him. They use the black star dragon balls to prevent the Earth's destruction after the fight.
Shadow Dragon Saga
At first everyone is happy that the threat is gone, tempting fate that "without Goku we'd be lost" or something like that. Just then Goku slowly begins to fade from sight, he is even confused about what's going on. He eventually disappears. They contact the King Yemma and the Kais about Goku's whereabouts but he isn't in otherworld, just "gone". They gather the dragon balls (Red Star of course) and try to wish him back from whereever he's gone. The dragon balls are cracked, they just assume that its a sign of them aging and summon Shenlong anyways. The dragon is twisted and mutated, crying out in pain. He begs the peeople to hurry up and make their wish so he could recuperate. They wish for Goku's return which he grants (at least so it seems). Goku is presented before them unconscious. Just then the dragon roars in pain and breaks into seven parts (which will be the different Shadow dragons). Each character (Gotenks, Gohan, Vegeta, Uub, Mr. Buu, and Pan) take on a different Shadow dragon which they all defeat. Syn Shenron absorbs the dragonballs and becomes Omage Shenron. The Omega Shenron fight will be in space, to avoid collateral damage. The warriors surround themselves with Ki (sort of like Green Lantern energy) so they can breathe. Omega easily beats all of them, until Goku arrives. He, with IT, appears behind Omega Shenron and fires a Kamehameha wave at him. In instead of damaging him the attack saps Shenron of his energy which Goku absorbs. Eventually Omega Shenron is reduced to dust and the dragon balls he absorbs are ordinary stones.
Dark Goku saga
After the Shadow dragons are eliminated, they all thank Goku for his help. Vegeta is bothered that Goku "just had" to save the day again. He retires to his training facility so he could study a way to compete with him again. Goku keeps away from other people (including his family) they wanted to ask him where he went when he disappeared but they could never catch him for more than a few moments. Goku joins the world Martial arts Tournament. Goku viciously breezes through each match, paralyzing or dismembering his opponents as they come (though not killing them nor giving them the opportunity to forfeit). Whenever people try to ask him if he's okay he just ignores them. Goku's last opponent is the champ himself, who expect Goku to go easy on him. Not only does Goku beat him but reveals that he's a fraud. The Z fighters (sorry for the dub name) tell him that he's gone too far, and his behavior is different. Sort of like Majin Vegeta, Goku kills the spectators. Biting off limbs eating skin etc.
I'll continue this later if anyone cares.