After going south through Route 12, you can continue on to Route 13 on your way to Fuchsia City.
Explore the Route
Go south from the east side of Route 13, and you can battle Camper Nash, who has a level 37 Sandshrew and a level 37 Sandslash. You get 740 Pokédollars and 2 Great Balls for winning.
To the left, you can battle Picnicker Susie, who has a level 37 Jigglypuff and a level 37 Weepinbell. You get 740 Pokédollars and 2 Great Balls for winning.
Go west from there. When the path splits, you can go north to battle Beauty Beauty Lola, who has a level 37 Chansey. You get 2220 Pokédollars and 3 Poké Balls for winning.
Go west from the Beauty, then go north after the wooden sign, then go east. Chop down the tree and you can battle Ace Trainer Kite, who has a level 39 Graveler, a level 39 Magneton, and a level 40 Haunter. You get 4000 Pokédollars and 5 Ultra Balls for winning.
In the grass to the east, you can encounter Pidgey, Pidgeotto, Farfetch'd, Weepinbell, Krabby, and others.
Go east through the grass to find a PP Up.
Go west, then south, then east. Go south from the Beauty, then west to battle Bug Catcher Milo, who has a level 36 Scyther and a level 36 Pinsir. You get 432 Pokédollars and 3 Poké Balls for winning.
Go southwest, then west, to battle Picnicker Gwen, who has a level 37 Raticate and a level 37 Golduck. You get 740 Pokédollars and 2 Great Balls for winning.
Southwest of Picnicker Gwen, there is a hidden Golden Pinap Berry.
Go north from the Picnicker, then west as far as you can go, then south, then west. Check the ground at the dead end to find a Golden Pinap Berry.
Go east, then north, then east, then north again, then east, and you can battle Beauty Sheila, who has a level 37 Eevee and a level 37 Persian. You get 2220 Pokédollars and 3 Poké Balls for winning.
Go north, then east from there, and you can battle Bird Keeper Robert, who has a level 39 Doduo and a level 39 Pidgeotto. You get 2340 Pokédollars and 3 Poké Balls for winning.
Go north from the Bird Keeper, then west, and you can battle Camper Perry, who has a level 37 Growlithe and a level 37 Nidorino. You get 740 Pokédollars and 2 Great Balls for winning.
Go southwest from the Camper and you can battle Punk Guy Jared, who has a level 38 Golbat. You get 1064 Pokédollars and 3 Poké Balls for winning.
Go north from the Punk Guy, then west, and you will be in Route 14.