You get two races in a row now, both on different vehicles :)
First, you're told to find artifacts in the desert, doing it quickly enough to avoid the storms that are coming.
Really, this is just a hoop race, moving from green spot to green spot with a timer running and people chasing you. There are about 20 or so of the target spots that will have you racing around the outer wasteland map. You use that wasteland map quite a bit in the game, so it's a good time to learn how it's laid out and where the major features are.
The last marker is back at the base again. "Good work", he says. Being the nice, friendly guy that he is, he lets you keep the armor you found.
Head on over to monk-girl. "This isn't a game" says the girl, he he he. She wants you to do a leaper race to prove your worth. It's just a big loop around - learn the course and then perfect your turns and jumps. There's no real trick to this one. When you do it, she gives you a light crystal. She also lets you in on the big plot motivator of this game - that the day star approaches. The whole "issue" in Jak 3 is that the star is coming and will destroy all life when it gets here.
Head on into the big door, but all the other carts are still locked against use. You'll unlock them slowly as the game progresses.
Jak and Daxter 3 Walkthrough