Mount Paozu (Return) Edit
After the Dragon League and the credits roll, you're brought back to Mount Paozu. Dr. Brief will bring you to his lab, taking your and your Rival's Scouters and updating their data, allowing you to find, catch, and evolve many more fighters than before. This is referred to as the National Scouter.
For now, you can't rematch the Dragon League since it's blocked off. Head to Celio on One Island to get your first post-game task.
The Ruby Edit
Celio will request that you find a Ruby on One Island for him, so head up to Mt. Ember and go right entrance. Two Pilaf Grunts will accidentally drop a password for their hideout, then challenge you. Beat them, and enter the cave. The path is straightforward, and you'll find the Ruby at the end of it in the basement. Head back to Celio with it.
The Sapphire Edit
After taking the Ruby, Celio will meekly ask for the Sapphire as well. This is a much longer quest than the first—in fact, he'll give you the Rainbow Pass to access Four through Seven Islands, as well as a whole new map page just for this purpose.