After you go through Route 116, you enter Rusturf Tunnel to find the Team Magma/Aqua Grunt who stole the Devon Goods and Peeko.
The only pokémon that you can encounter in the cave is Whismur.
Get an Item
Go north from the cave entrance and you will see an item on the ledge. It's a Poké Ball.
Battle the Grunt
Go east to battle the grunt. The Team Magma/Aqua Grunt has a level 11 Poochyena. After you win, you get the Devon Goods. Mr. Briney will come get Peeko.
Go back Outside
There is a dead end to the east. Later you can teach your pokémon a move that will give you access to the rest of the cave. For now, go west and go back outside.
Return the Stolen Goods
Go back to Rustboro City and give the Devon Worker the stolen goods. He gives you a Great Ball. He lets you visit the Devon Corporation and he asks you to deliver the goods to the Shipyard in Slateport. Then he asks you to meet with the president of Devon Corporation. Mr. Stone asks you to deliver a letter to Steven in Dewford on your way to the Shipyard. In exchange, he gives you the PokéNav. It shows you the map, the condition of your pokémon, and a list of some of the trainers that you have battled before.
Go to Mr. Briney's Cottage
Go outside and go to the south exit of town. You will see your rival standing there. In Emerald, you can add your rival to your PokéNav and optionally have a battle now.
Next, you have to get to Dewford Town, but as you can see on the Hoenn Map, you have to take a boat. Mr. Briney can help. His house is to the south. Go south through Route 103 and the Petalburg Woods. Mr. Briney's cottage is there. Go inside and talk to Mr. Briney. He agrees to take you to Dewford Town.