The underground path opens up into Route 6. There is another guard house to the north, but you'll encounter the same problem the first time you get here: the guard's too thirsty and won't let you pass. So for now, head south into Vermilion City.
The trainers on this route can be avoided. If you don't feel like fighting, stay out of their line of sight. However, you can't pass the last two without a battle.
Bug Catcher Keigo Weedle (16) Caterpie (16) Weedle (16)
Camper Ricky Squirtle (20)
Picknicker Nancy Rattata (16) Pikachu (20)
Bug Catcher Elijah Butterfree (20)
Picknicker Isabelle Pidgey (16) Pidgey (16) Pidgey (16)
Camper Jeff Spearow (16) Raticate (16)
Duck Pond
After you obtain the HM needed for Surf in the Safari Zon, try swimming around the pond in the south of Route 6. This pond is populated by Psyduck, as well as the occasional Golduck. Remember, a Golduck evolved from a high-level Psyduck tends to be stronger than one you caught in the wild.