Once inside Viridian city, heal your pokémon at the Poké Center. Then go to the Poké Mart. Before you can buy anything you must take Oak's Parcel from the cashier and take it back to Prof. Oak in Pallet Town.
Go back to his lab and give him the parcel. In return he will give you and your rival a Pokédex to keep up with the pokémon you have caught. Now, go back to Viridian City and to the mart. Buy whatever you want for your journey and then go North-West. Talk to the guy who is walking around and if you say NO to his question he will teach you how to catch Weedle. Enter route 2. In Route 2 bring your pokémon's levels up a couple levels and go into the house ahead, through the door and into Viridian Forest.
Viridian Forest:
This forest contains 4 or 5 different pokémon. Once of them is a pikachu which I recommend getting. Search the forest for a poké ball, antidote and potion. Go far left and up to get to another building. Go through the building and through the field untill you get to Pewter City.