Pokémon Red and Blue/Route 2 & 22

'These routes are a good place to collect more Pokémon. Route #22 (shown below) is full of, along with a specimen or two that are rarely found in this area. Remember to weaken a Pokémon in battle before using a Poké Ball on it, or else it will be able to break free. The first time you come through Route #2, head into Viridian Forest to continue your journey.'

Your rival awaits
It's a good idea to take some time and train your Pokémon on Route #22. You will have the chance to catch a Mankey, but they are only in Yellow version. Be sure that you do if you're playing it. Otherwise, be sure to pick up whichever Nidoran is more common in the version you're playing, both if you're trying to complete your Pokédex. If you stumble upon a Spearow, it makes a nice addition to your collection, being a better Flying-type than Pidgey.

If you would like to earn extra EXP (and 280), you can challenge your rival here by heading west towards the pond along the top path. Before you do, make sure that you have a few strong Pokémon, and that they're healthy and ready to fight. You may wish to wait until after you've traveled through Viridian Forest to conduct this battle or you may want to skip it entirely.

If you beat your rival, you can head back to Pallet Town to get 5 Poké Balls for free from Professor Oak in his lab (but only if you have not purchased any Poké Balls yet, so it's really not worth the hassle of fighting your rival with only your starter Pokémon).

When you're done at Route #22, make your way back to Viridian City and exit to the north to reach Route #2. Much of this route is inaccessible to you at this point in the game, but that will change in time. For now, your only concern is reaching the entrance to Viridian Forest. Don't bother hunting for Caterpie or Weedle out here; they're plentiful in the forest.

Rival battle #2
Your rival has improved somewhat since Pallet Town. In Red and Blue, he's using a Pidgey in addition to his starter; in Yellow, a Spearow. You will also want to have some new Pokémon at your disposal. The strategy below is not essential, but it makes things much simpler.

The first order of business is acquiring a Spearow, since it's right here on Route 22. Anyone not using Charmander should probably acquire a Spearow. Training it to Level 7-8 will be sufficient. After that, go to Viridian Forest and capture a Pikachu. You'll want to do this no matter what starter you have, because Pikachu can counter your rival's new Pidgey (as well as two out of his three starters). Level Pikachu up to Level 9 for Thunder Wave.

Your rival always leads with Pidgey, so have Pikachu in the lead and K.O. it with 2-3 Thundershocks (preferably two). If your rival chose Bulbasaur, switch to Spearow and use Peck until Bulbasaur faints; if he chose Squirtle, keep Pikachu in and use Thundershock repeatedly, sending in Spearow if Pikachu faints. Charmander has no true counter yet, so you'll have to rely on a little more strategy, namely Pikachu's Thunder Wave. Paralyze, then attack until one Pokémon faints. If Pikachu goes down first, Spearow can finish it with ease.
 * Battle strategy

In Yellow version, your rival has Spearow and Eevee. Pikachu will destroy Spearow without any difficulty whatsoever, and Eevee is best handled by catching a Mankey on Route 22 and raising it to Level 9 for Low Kick. Pikachu's Thunder Wave, when Pikachu learns it at level 8, also helps against your rival's Pokémon.