The best way is through the Dream World. After logging in, you can either:
- Go to the Island of Dreams and look around.
- Go to other peoples homes and look at their Share Shelves.
Once you have a Berry you want more of, plant it in the Dream World.
These are the only ways to get them in-game:
- Wild Pokemon
- Event Pokemon (depending)
- Gift from NPCs
- Flower Shop at Join Avenue (Credit to: StarsignOfTheAbyss)