Can we talk about how great a character Vegeta is?
I personally feel that Vegeta was one of the most tragic characters in the whole series.
Goku got to grow up and live a nice life on Earth. Fall in love with his childhood sweetheart, had kids, great friends, and was loved by everyone.
While yeah, Vegeta does have a wife, kids, and friends, his past definitely influenced his present self a lot. Vegeta always talks about his pride as a Saiyan and how strong they are. I personally feel that he deeply cares for his friends just as much as his family.
If we just look at the surface of it, he's just someone who keeps bragging and boasting. Maybe someone already said it before me, but I feel that it's much more than that. He lost his entire race to Frieza, and ends up having to work for him or face death.
I feel that, when he talks about his pride as a Saiyan, he truly wants the Saiyan race to be remembered, not just how strong they were. If you watched Dragon Ball Super, the arc where he meets Universe 6 is a great example. When he meets Cabba and learns about their surviving race in that universe, a part of him wants to meet everyone. He probably wants to see the Saiyans again, even if they were not from the same universe. Vegeta teaches Cabba SSJ to ensure that he and other Saiyans are strong enough to defend their race for when the time comes, unlike how SSJ was a myth that only legendary Saiyans could accomplish, resulting in their mass genocide to FrIeza.