Pallet Town

Walkthrough
After finishing the first battle against your Rival, the goal will be to head north of Pallet Town on Route 1 towards Viridian City. As for Pallet Town, there isn't much to it at this point in the game. You won't be able to surf or fish in the water to the south of town until later in the game, so really the only thing you can do before heading to Route 1 is obtain the Town Map from your Rival's sister in their home.

Wild Pokemon
Surfing:
Tentacool (100%): Levels 5-40

Old Rod:
Magikarp (100%): Level 5

Good Rod:
Horsea(FireRed)/Krabby(LeafGreen) (60%): Levels 5-15
Krabby(FireRed)/Horsea(LeafGreen) (20%): Levels 5-15
Magikarp (20%): Levels 5-15

Super Rod:
Shellder(FireRed)/Staryu(LeafGreen) (40%): Levels 15-25
Horsea(FireRed)/Krabby(LeafGreen) (40%): Levels 15-25
Gyarados (15%): Levels 15-25
Seadra(FireRed)/Kingler(LeafGreen) (4%): Levels 25-35
Psyduck(FireRed)/Slowpoke(LeafGreen) (1%): Levels 25-35

Items
Potion - From Home PC
Town Map - From Rival's Sister