The first building you'll reach is the Pokémon Center. This Pokémon hospital will heal your team to full health and normal status, and Trainers are free to use the PC in the corner. In addition to accessing your item storage, these PCs can also access your Pokémon storage. Since you can only carry six with you at once, any others are immediately transferred to storage upon capture. It is typically a good idea to heal up and save your progress whenever you enter a Pokémon Center.
Behind the Pokémon Center is the local Poké Mart. These stores offer a variety of Poké Balls, Potions, and other items. When you step inside, the cashier asks you to deliver a parcel to Professor Oak. The cashier will not sell anything until the delivery is made.
When you deliver the parcel to the professor, his grandson runs in and starts yelling about how much his Pokémon has grown. Oak gives you both a Pokédex, a high-tech Pokémon encyclopedia. This little machine will record all available data on the Pokémon you encounter, and will be complete when you catch a member of every species. Visit your rival's sister Daisy before leaving town to get a Town Map.
The old man that had blocked your way northward has had his coffee now. When you pass him, he tries to show you how to catch a Pokémon. It doesn't work though, and he runs off to buy more Poké Balls. Before heading north, detour to Route 22 in order to battle your rival and catch some new Pokémon.
Route 22 is a short route that runs west of Viridian. The path is a dead-end right now, but you can battle your rival and hunt for new Pokémon here.
Keep in mind that your rival's team later in the game is dependent upon the outcome of this battle along with the previous one. If you want your rival's Eevee to evolve into an Electric type, you had to have won the first battle and win this battle as well. If you want your rival's Eevee to evolve into a Water type, you had to have lost the first battle. If you want your rival's Eevee to evolve into a Fire type, you had to win the first battle and either lose or skip this battle.