Goku had avoided Freeza?

[Hector]

Let's say he had decided to listen to Kaio's advice and just take his friends and leave Namek instead of messing with Freeza.

Here is what I think would have happened...

Goku offers to take Vegeta to Earth too, which he accepts, since, in canon, when he thought that Freeza had obtained immortality, he said that the best tactic is to wish they never come across him again. Freeza does not manage to get his wish, destroys the planet and leaves.

On Earth, Vegeta stays at Bulma's and trains to surpass Kakarrot as in canon. Alternate Trunks appears and reveals to them the Super Saiyan form. Over the following 3 years, both Goku and Vegeta acquire the form.

3 years later, all Z fighters remaining alive are as strong as in canon (well, Goku/Vegeta were minus one zenkai, but I bet they got it during the 3 year training). The first part of the saga goes as in canon, with Krillin being yamchaed by 20 in place of Yamcha and Vegeta fighting Gero in place of Piccolo later. With no Kamiccolo, Cell remains hidden.

Krillin is the only one left to defend the Kame House. Goku tells Gohan to stay at Kami's lookout and teleports to save his friend. He starts fighting Seventeen and is losing when Cell shows up. The fight goes as in canon, with Goku replacing Kamiccolo and getting killed. With no one left to stop him, Cell has become perfect by the time Vegeta/Trunks exit the room.

After defeating Vegeta/Trunks, Cell announces the Cell Games. Both Vegeta and Trunks go in the ROSAT a second time (Trunks goes with Gohan), but Cell still kills everyone in the end.

 

[theallpowerfulpuipui]

Cell would never exist but the 'droids stomp.

 

[Captain Cadaver]

I don't think Vegeta would go to Earth. Not only would Kuririn not omit the detail on Namek's sky not going dark, but he'd have no reason to live on Earth when Goku hadn't proven himself to be a Super Saiyan yet. I think Vegeta would remain on Namek in an attempt to get his wish, possibly killing Ginyu if biding his time, but being killed by Freeza regardless of his efforts. Freeza would likely still go to Earth after this due to the Earthlings having interfered with his plans significantly, which would lead to him destroying it.
On the small possibility he decides not to bother, things go similarly to Trunks' timeline except that Bulma starts work on the Time Machine a lot sooner and Gohan goes back in time to when Nappa and Vegeta arrived, killing them before starting work on finding Gero's lab with the help of everyone. With the Spy Robot though, Gero would be aware of this and make whatever contingencies he had to, such as hiding in his underground lab. Whether or not they find him, the Dragon Team would all be killed by Dabura about a decade later.

 

[Hector]

I don't think Vegeta would go to Earth. Not only would Kuririn not omit the detail on Namek's sky not going dark, but he'd have no reason to live on Earth when Goku hadn't proven himself to be a Super Saiyan yet. I think Vegeta would remain on Namek in an attempt to get his wish, possibly killing Ginyu if biding his time, but being killed by Freeza regardless of his efforts. Freeza would likely still go to Earth after this due to the Earthlings having interfered with his plans significantly, which would lead to him destroying it.
On the small possibility he decides not to bother, things go similarly to Trunks' timeline except that Bulma starts work on the Time Machine a lot sooner and Gohan goes back in time to when Nappa and Vegeta arrived, killing them before starting work on finding Gero's lab with the help of everyone. With the Spy Robot though, Gero would be aware of this and make whatever contingencies he had to, such as hiding in his underground lab. Whether or not they find him, the Dragon Team would all be killed by Dabura about a decade later.

I don't see why Bulma works on the time machine sooner or why Gohan goes further back in time. Also, most likely, Gohan is killed by the Androids all the same and Bulma goes back in time instead.

 

[Warmmedown]

Vegeta would be totally different. If he did go to Earth, he wouldn't be as amicable, because he wouldn't have gone through his death and talk with Goku, which I think likely gave him more respect and affinity towards Goku and generally opened the door for him to soften up a little. Maybe Super Saiyan could still occur on Earth if Vegeta is around, because there's a chance he would kill someone and send Goku SSJ, but it's also possible they just wouldn't get strong enough. Whether Bulma would live with Vegeta I'm not sure, since he wouldn't be considered as close to being one of the gang if he hadn't fought alongside Krillin and Gohan against Freeza.

 

[Captain Cadaver]

I don't see why Bulma works on the time machine sooner or why Gohan goes further back in time.

Because Gohan would be the only hope left rather than having a backup in Trunks, pressuring Bulma to work on the time machine sooner, and the consequences of the Saiyan fight (many characters being killed off due to Piccolo's permanent death) would make it the most crucial point to travel back to for an ideal future.

Also, most likely, Gohan is killed by the Androids all the same and Bulma goes back in time instead.

As reckless as he was, it still took give or take two decades for him to slip up like that. With the scenario being one that would logically have Bulma try the time machine sooner, there's no reason to believe Gohan would die before its completion.