Route 103 is located at the north of Oldale Town and west of Route 110. If you can't find Prof. Birch in his lab, he is probably here to do his fieldwork. After you become the champion of the Pokémon League, a new cave Altering Cave will be unlocked (Pokémon Emerald only) and will be next to the water bodies (east).
Prof. Birch's child is waiting for a battle in Route 103. This will be the first battle for you against a trainer. This is the first battle between you and your rival.
Prof. Birch's child uses a different Pokémon in the battle depending on which Pokémon you selected from the Professor after the rescue in Littleroot Town. If you choose Torchic, look out for Mudkip. If you choose Mudkip, look out for Treecko. If you choose Treecko, look out for Torchic.