After you receive the Boulder Badge, other Pokémon trainers will see you as a worthy opponent. No fewer than eight trainers wait to challenge you along Route #3. You'll have a tough time getting through them all, but you'll emerge from Route #3 a more powerful (and richer) trainer than ever. Stock up on Poké Balls to capture some more Pokémon in the wild too.
On the way to Mt. Moon
There are eight Pokémon trainers between Pewter City and the entrance to Mt. Moon, but they're pretty easy to avoid. Some of them can pose a threat since they are close to the same level as your Pokémon. Still, you may need the EXP.
Mt. Moon is tough because you will run into many wild Pokémon, as well as trainers who use a mix of a few Pokémon types. This is a good place to get a Pidgey or Spearow for your Flying-type, having a 90% chance (45% chance each) of getting one of them in Red and Blue and a 55% chance of getting a Spearow in Yellow.
|#39 Jigglypuff||#27 Sandshrew|
BUT if you can get one to level 20, it will evolve into Gyarados, a Water/Flying-type with great stats (with a great Attack and good Special stats) and a small but useful move pool that includes the powerful Hydro Pump and Hyper Beam (and several more great moves by TM).
If you want, you can buy it to begin leveling it up now. However, this will devour many experience points that would have (and should have) gone to other Pokémon at this early stage. You can fish for a Magikarp anywhere some other time once you get the Old Rod, and it's probably a good idea to do so. Then, train it up against more higher level Pokémon by sending Magikarp as the first Pokémon, then switching to one of your stronger Pokémon to finish off the opponent.