After you go through the crater-covered Route 114, you see your neighbor rushing after Team Magma/Aqua, who went to Meteor Falls.
Explore the Caves
In here, you can encounter Solrock (in Omega Ruby) or Lunatone (in Alpha Sapphire) and Zubat. If you use the Old Rod here, you can catch Goldeen and Magikarp.
Go north from the entrance to find a Full Heal.
Go back to the south and cross the bridge. You will find your neighbor demanding that Team Magma/Aqua let Professor Cozmo go. Your neighbor asks you to battle together.
In Omega Ruby, you will be up against Team Magma Admin Tabitha, who has Mightyena and Numel, and a Team Magma Grunt, who has Koffing.
In Alpha Sapphire, you will battle Team Aqua Admin Shelly, who has Carvanha and Mightyena, and a Team Aqua Grunt, who has Grimer.
After the battle, you and your neighbor will take Professor Cozmo back to his lab in Fallarbor. The professor gives you TM23 Smack Down.
Afterward, your neighbor will ask if you want to go to Mauville together. If you say yes, you will go there instantly.
If you say no, you will stay here in Fallarbor. You can get a Moon Stone in Meteor Falls and go through the cave to reach Route 115, which leads to Rustboro City. From there, you can go to Rusturf Tunnel, and from there to Verdanturf Town, which leads to Route 117. From there you can get to Mauville City.
In Mauville City, go to the cycle shop and switch to the Acro Bike, which will be helpful soon. Then go north from Mauville City to reach the cable car to Mt. Chimney, which Team Magma/Aqua are no longer blocking. Before using the cable car, go to the right of it to find a Guard Spec. Then ride the cable car to Mt. Chimney.