Infestation! Saibamen Scourge
Win: Defeat 20 enemies
Lose: Player's HP is depleted
For the first part, you have to defeat three Saibamen. After dealing with them, you'll be surrounded by several more, including the more powerful Grey ones. The mission will be automatically completed after you defeat 20 of them, including the Grey Saibamen. After completing this objective, you have the choice to continue to the next scenario or return to the Time Nest. If you chose the latter, simply talk to Trunks to resume.
Arrival of the Saiyan Warriors
Win: Defeat all enemies
Lose: Player's HP, Gohan, or Krillin's HP is depleted
First you have to battle Nappa with Piccolo. As you deal damage to him and wear him out, a cutscene will trigger and Piccolo will sacrifice himself to cover Gohan. In the second phase of the battle, you'll fight him with Gohan and Krillin. Further lowering his HP gauge will trigger yet another cutscene where Goku arrives. This time, you'll have to deal with Nappa alone while Goku handles Vegeta. Be careful with Nappa if you're a striker since he seems to love to grab opponents. After taking him out, join Goku and defeat Vegeta.
Power x10! Vegeta Transforms!
Win: Defeat Great Ape Vegeta
Lose: Player and Ally HP depleted
This battle should be easy enough. First, you have to keep attacking the Great Ape to drain its stamina and force it to a "Break State". You can also grab its tail to perform a throw attack that will greatly reduce its stamina. Executing a throw while the Great Ape is in "Break State" will deal huge damage to it and also give you a chance for a follow up attack. In the first phase, just attack it or aim for its tail. Goku should be able to completely deplete the first life gauge by lifting the Great Ape by the tail and slamming it to the ground.
After the cutscene, carefully attack it and keep an eye on its attack. Fully depleting the stamina will trigger a cutscene with Krillin successfully cutting off the Great Ape's tail and making him lose power. scenario complete!
Prince's Pride! Limit Battle
Win: Defeat Vegeta
Lose: Player's HP depleted
This battle should be manageable with Goku's help. However, don't underestimate Vegeta. Even when critically wounded, he can still put up a fight. Just perform your combos and be aggressive to build your gauges so you can perform more potent finishers.