Vegeta Missions don’t unlock until after the Frieza Saga. I was doing Instructor training with Nappa and his first mission randomly popped up.
Sparring With Vegeta Easy – By the time you get this mission, you should have no trouble defeating Vegeta. Defeating him will net you the Training Suit(CC), increases Ki attacks but lowers Ki overall.
Sparring With Vegeta Intermediate – I did this after the Future Saga so I was way over powered for this point. Vegeta hits hard and is in Super Saiyan for this fight, be ready. Winning will get you the Gi(CC).
Sparring With Vegeta Advanced – This one can be a bit challenging so bring a couple healing items. I think he is in Super Saiyan 2 during the fight but it is hard to tell, I just know he hits hard. Winning here gets you the Elite Suit(CC).
After you beat Vegeta all three times you get a trophy and a Distorted Time Egg, talk to Bulma for the egg.
The Final Showdown With Vegeta – This is it, Super Saiyan God Vegeta time. This is easily the most difficult mission that Vegeta has. The reward though, is huge, Super Vegeta. You also unlock the Hero Suit(CC) for your trouble.