After you get through the long grass of Route 119, you find some Team Aqua/Magma Grunts blocking the bridge, so you enter the Weather Institute.
Rest Up
There is an area in the northwest of the building with a bed and a PC. You can check on the bed to rest and heal your pokémon. You can use the PC to switch pokémon in your party.
Defeat Team Aqua/Magma
Go east from there and battle the Team Aqua/Magma Grunts. The first Grunt, who is sitting at a computer, has a Numel or Carvanha. The second Grunt has Zubat and Poochyena.
Go upstairs. The Grunt to the right of the desks has Poochyena and Numel/Carvanha. The Grunt to the left has Poochyena, Zubat, and Numel/Carvanha.
To the left, you have to fight Magma Admin Courtney or Aqua Admin Shelly. Courtney has a level 28 Numel and level 28 Mightyena. Shelly has level 28 Carvanha and level 28 Mightyena.
Receive Castform
After you defeat the Aqua/Magma Admin, the nearby scientist gives you a pokémon. If you don't have room, use the PC downstairs next to the bed to deposit a pokémon. The scientist gives you Castform. It can change form depending on the weather. The Castform is holding a Mystic Water.
After receiving Castform, go downstairs and heal up in the bed if needed. Then go back outside.
Continue on Route 119
Go east across the bridge. On the other side, you can go east and Surf south to find a Leaf Stone, which can evolve certain pokémon.
Go north from the bridge and your neighbor will challenge you to a battle.
Neighbor Battle
In Ruby and Sapphire, if you chose Treecko, your neighbor has level 29 Shroomish, level 29 Wailmer, and level 31 Combusken. In Ruby and Sapphire, if you chose Torchic, your neighbor has level 29 Shroomish, level 29 Numel, and level 31 Marshtomp. In Ruby and Sapphire, if you chose Mudkip, your neighbor has level 29 Wailmer, level 29 Numel, and level 31 Grovyle.
In Emerald, if you chose Treecko, your neighbor has level 29 Lombre, level 29 Pelipper, and level 31 Combusken. In Emerald, if you chose Torchic, your neighbor has level 29 Lombre, level 29 Slugma, and level 31 Marshtomp. In Emerald, if you chose Mudkip, your neighbor has level 29 Slugma, level 29 Pelipper, and level 31 Grovyle.
After the battle, your neighbor gives you HM02 Fly. You can't use it outside of battle yet, but you will after you get the next gym badge.
Go north and you will pass by some fake trees. If you check on them, a Ninja Boy will attack. There are some Berry trees to the northwest. Go east to reach Fortree City.