After you rescue Mr. Briney's Peeko in Rusturf Tunnel and retrieve the stolen Devon Parts, Mr. Briney takes you to Dewford Town.
Explore the Town
North of the Pokémon Center, you find a kid who talks about what's trendy. If you say that the thing that he's talking about isn't trendy, he asks you what is. You can choose an item from your Bag (you keep this item) and then you can choose a word. The item and the word that you chose becomes the new trendy thing. This will affect the game later, but you can always change it by talking to this kid.
Talk to the fisherman near the Dewford Gym to get an Old Rod. Now you can catch pokémon in the water! In Dewford Town, you can catch Tentacool and Magikarp with the Old Rod.
In the northeast house, talk to the Ace Trainer to get a Silk Scarf. It boosts the power of Normal-type moves.
If you want to do some training before you go to the Dewford Gym, you can go northwest to Route 106, then west to Granite Cave, where you can catch Zubat, Geodude, Abra, and Makuhita. The hiker inside of the cave gives you TM70 Flash.
You can't go to the inner part of the cave, though, because some ruin enthusiasts are blocking the door, saying something about a mural in the cave of a huge pokémon that nobody has seen before. So you will have to leave and come back later.
You could also go east from Dewford Town to reach Route 107, where you find a Super Repel.
When you're ready, you can go to the Dewford Gym.