* Desert City
Found easily on the initial city rat-finding mission, 8 gems gets you a ring challenge. Getting each ring gives you more time to find the next. hop on leaper and go through them. It will probably take you a few laps to learn the course. Be sure to use the stairs you come across. You hit 13 at the bottom of the red stairs; there are 16 or so total. When you win, you get eggs with which to buy secrets.

* Time Freeze Trial
In this one, each ring makes time stop running forward. You have to learn which rings are worth going out of your way to get - some take longer to get to than they give you in return.

Jak and Daxter 3 Walkthrough