Viridian Forest is a very small area within Route 2. You’ll come across a few trainers as you travel through, so make sure your team is in top condition after each battle. You can always run back to the Pokemon Center in Viridian City to heal your team for free, but it’s easier to buy some Potions and Antidotes to bring along. Also, make sure you have some Poke Balls to capture a few Pokemon in the area. Getting a Caterpie here and evolving it to Butterfree is a great option for the upcoming gym battle.
The first thing you should do is travel to the west. You’ll run into a trainer in the tall grass with two Nidoran.