Route 1 is a very short route north of Pallet Town. There are no trainers, but there is a lot of tall grass with wild Pokemon throughout the route. You can level up your starter Pokemon a little here before going on. If your Pokemon gets too weak, you can return to Pallet Town and heal them by speaking to your mother.
Unfortunately, you won’t be able to catch any Pokemon just yet because you don’t have any Poke Balls. Follow the path north until you reach Viridian City.