If you want to train your partner, or any other Pokémon, after you beat Rayquaza, you can do one of two things:
- You can take them with you on missions. To do this, you go to their Friend Area, click on/tap on/select them and then choose "Join Team".
- Or you can train them up individually by making them the leader. To do this, follow the same steps as above. Then select them again once they're in your team and select "Make Leader". That Pokémon will then become the leader, as you can imagine.
From then on, you can either send the previous leader back to their Friend Area and go alone, or you can let them stay and go on missions with you. It's personal preference, really.