Saiyan Arc:
-Yamcha, Puar, Oolong, Tenshinhan, Chaotzu, and Launch are all at Kamesennin's place when Goku visits. The whole scene plays out more or less the same, though. Maybe the other Z-senshi help Goku and Piccolo fight Raditz, but Raditz plants some saibamen to keep them busy while he fights Goku and Piccolo. The end result is the same: Goku dies, Piccolo takes Gohan, everyone goes to Kamisama's for special training, etc.
-The power creep is kept to a minimum. I don't like how all of the human Z-senshi are suddenly stronger than Raditz after just a year. Instead, Raditz is considered to be really strong and is still much stronger than any of them even after their training. Only Piccolo and Gohan could take Raditz one-on-one, and it'd still be a deadly battle. Maybe something like Tenshinhan's Kikoho could injure Raditz, but none of them could just fight him by themselves. Nappa is stronger than Raditz by a decent margin, but he's not multiple times stronger or anything.
-I think there might be a female Saiyan that shows up with Vegeta and Nappa, who's about equal with Raditz (and hence, Piccolo). I don't know how the whole Saiyan battle would play out, but she'd survive the battle and leave the planet with Vegeta. Yamcha, Chaotzu, Tenshinhan, and Piccolo would still end up dead. I think maybe Piccolo and the female Saiyan would go one-on-one while everyone zerg rushes Nappa. Piccolo uses the Makkankusanpou to tear a hole through her shoulder and she falls under some rubble and is out like a light, then Piccolo starts helping the others fight Nappa but Chaotzu, Yamcha, and Tenshinhan are all dead and he's exhausted from his own fight. The rest probably plays out like it did originally until Goku shows up. Later, while Goku's fighting Vegeta, the female Saiyan wakes up and maybe fights Kuririn or whatever. Regardless, the end result is the same: Vegeta and her both leave in defeat and limp off back to Freeza's planet.
-Maybe Chi Chi would also train to prepare for the Saiyans and fight with the other Z-Senshi, and survives the battle. Launch, meanwhile, is back at Kame House with the others, watching the battle on TV or with Uranai Baba's crystal ball.
Freeza Arc:
This is more complicated. Okay, first, I'd have Goku be given a senzu bean so that he heals right away after his fight with Vegeta. Then he goes with Gohan, Kuririn, and Bulma to Planet Namek. Maybe Launch and Chi Chi also go. I like the idea of Launch sticking around during DBZ, rather than just disappearing. She would totally have gone to Planet Namek to wish back Tenshinhan! Among these changes, I would have:
-Ditched Kui and probably Dodoria. Instead, it's just Zarbon who's by Freeza's side. I'd also change the power scaling, so that "pretty" Zarbon is a little weaker than Raditz, but his monstrous form is about equal with Nappa. Instead of Kui, it's Commander Ginyu who's Vegeta's rival and equal. They're about equal in power, and each of the other Tokusentai (save Guldo) are about equal with Nappa and monster-Zarbon. Then a little below them are Raditz and the female Saiyan, then "pretty" Zarbon. Then there's a huge gap until you get to the rest of Freeza's army. So Raditz and Nappa are notoriously strong in the grand scheme of things. I don't like how Raditz is only supposed to be as strong as a rank-and-file soldier. Here, he's an elite. The rank-and-file of Freeza's army are weak even by Dragon Ball standards, as is Guldo, and probably aren't even much stronger than Red Ribbon soldiers. There's no way that Appule is stronger than Tao Pai Pai. Maybe there are stronger soldiers who, instead of being rank-and-file are arranged into teams that attack planets in those little ball ships, but even those are probably equal to Tambourine, Drum, etc.
-Obviously with Goku being on Planet Namek from the very start, no Kui or Dodoria, Vegeta already being as strong as Ginyu, the female Saiyan (dont have a name for her right now) going with Vegeta to Planet Namek etc. the story would play out differently. I don't feel like writing a complete fanfiction right here and now, but the basic gist is that Goku and Gohan get their potentials unlocked by Saichouro. Somewhere along the way, Piccolo gets wishes to Planet Namek and gets his potential unlocked, but never fuses with Nail. Vegeta and the female Saiyan get their potentials unlocked when it becomes clear that they're going to be needed to fight Freeza. There are no zenkai power-ups. Kuririn gets his potential unlocked, but is still short of Raditz. Tenshinhan and Yamcha are also about equal with him by the end of the arc, too, with Chaotzu being significantly behind but a bit stronger than he was when he fought the Saiyans.
-I would either just get rid of the whole Super Saiyan concept to begin with, or drastically change how it works. I think I might make it like the Super Saiyan God in Battle of Gods, where the others sort of contribute their power to him and he transforms. Or maybe Goku can only unlock its power when he's super pissed off and has a warrior's fury and bloodlust, and the minute his anger goes away he reverts back to normal. Plus, there can only be one Super Saiyan. The point is, there should be limitations on it so that things like the Kaio-Ken are still useful in certain situations even after the Freeza Arc. Also, because the power of the Super Saiyan can only be unlocked when a Saiyan is almost insane with rage and bloodlust for a hated enemy, Goku can't use it against friends or during friendly spars. Only during the final epic battles with big bad guys can he turn into a Super Saiyan.
-Get rid of all of Freeza's different forms, and just give him maybe two forms. His first one looks like a cross between his first and his forth, and then his second looks like a cross between his second and his third, and is bigger than King Cold. In his first form, there's a decent-but-not-insurmountable gap between him and Ginyu/Vegeta, but his second form is way WAY stronger. After all of their Saichouro power-ups, Goku, Gohan, Vegeta, and Piccolo are each a little stronger than Freeza's first form, but in his second form, Freeza clowns them all. I won't go into how the entire fight plays out, but maybe at one point Vegeta grows his tail back and becomes an Ozaru, and in that form he can exceed 2nd form Freeza in sheer power, but Freeza moves too fast and cuts off his tail, then kills him. Goku, even at maximum Kaio-Ken, can't keep up, and then Freeza kills Kuririn. Goku explodes with fury, unlocks the Super Saiyan power, and he and Freeza square off while everyone else escapes. Maybe Freeza grows really big like Piccolo did at the 23rd Tenkaichi Budokai, and that's his ultimate final form, the form he was in when he blew up Planet Vegeta. I like the idea of a kaiju Freeza, but I'm probably the only one,

That's just a sort of basic gist of what I would do the Saiyan and Freeza Arcs. The Cell and Buu Arcs would play out differently based on this as well, but I won't go into those, save only to point out that the jinzoningen obviously wouldn't be stronger than Freeza (and in fact I'd probably have them be on par with Nappa and the Tokusentai, or Vegeta when he came to Earth), there wouldn't be so many Super Saiyans, and the human Z-senshi would still be semi-relevant.