Here is the thing about Goku, he loves fighting and prefers fighting different people more than fighting the same person again and again unless they have a score to settle, which is why he changes rivals from Tien to Piccolo to Vegeta(and even Hit in the DBS anime), he doesn't have an eternal rival but a rival of the month you see, thats his type. Althought marketing and Vegeta's popularity would tell you otherwise.
The closest thing to a rivalry thats not one sided with Vegeta was during Namek, Goku wanted to beat him 1vs1 since their last fight wasn't an 1vs1, meanwhile Vegeta while he did want to kill Goku for what he did to him, he was fine if somebody else killed him(which is why he left him with Ginyu), because at this point in the series Vegeta cared more about the end result rather than how.
After Namek Goku stopped caring about Vegeta as a goal since there were bigger fish in the pond, but now Vegeta's character resolves around Goku so he does care about him being the one to defeat him, this continues to the Buu arc, so now for years we are left with a rivalry in which is one sided.
This kind of rivalry also keeps going to the DBS anime(just in a friendly way instead of Vegeta's previous toxic one) but as you can see with Hit, Goku is fine with moving on from Vegeta, he has been for years but status quo due to Vegeta's relevance as the deuteragonist keeps Goku shackled from truly moving on.
In the DBS manga their relationship is written more like a generic rivalry so they both care about outdoing each other specifically though.