This tunnel links Vermilion City with Pewter City and is your only way of going backwards, at least until you can find a way to fly or swim. If you didn’t follow Misty, you can reach here by going east from Vermilion City and entering the cave just as you step onto Route 11.
Inside, head past the man in the nearby corner and down the ladders. You’ll emerge in the main part of Diglett’s Cave: a passageway that continues in a north-westerly direction. Follow the cave all the way, being wary of the Diglett and Dugtrio burrowing around; feel free to catch some for your Pokédex.
As far as we can tell, there aren’t any items to be found in the tunnel, hidden or otherwise. Which is somewhat surprising, but presumably few people come here to drop items. So don’t worry and keep going until the very end. Climb up the ladders that await you at the furthest corner, then exit the cave via the south.
Back to Route 2
Coming from Diglett’s Cave, you’ll arrive towards the north-east of Route 2. From here, if you slice down the skinny tree on the left with Chop Down, you can head north back into Pewter City. In fact, if you’ve got nothing better to do, you might want to go there now. Once in Pewter City, go over the north-east corner.
Towards the left is another skinny tree that can be removed with Chop Down. Past that, you can enter the Pewter City Museum via the side door, which will take you behind the reception counter.
Back at the Diglett’s Cave entrance, go anti-clockwise around the entrance, via the narrow path. Behind the entrance, you can bag yourself 3 Great Balls. Now head south from the cave entrance.
Use Chop Down to get rid of the skinny tree blocking the way, then keep going south towards the eastern gatehouse. Exit this gatehouse via the south, then press on towards the south-east corner. Along the way, you can pick up a Leaf Stone, which can evolve Oddish or Bellsprout, and a Super Potion from the floor.
If you’ve got time for a short detour, use Chop Down to slice one of the nearby skinny trees on the left. Then proceed south back to Viridian City. Once there, head over to the south-west corner of the city. There will be another skinny tree that you can hack down. On the other side, chat to the lonely fellow to receive TM11 Will-O-Wisp. Then return to where you left off at Route 2.
Once you’re ready to progress, go back and enter the researcher’s house. He’ll teach you another Secret Technique for your partner Pokémon: Light Up. As you just saw, it can be used to light up extremely dark places. The next place you’re heading to will be one such place (and the only place in fact).
Outside, you’ll immediately bump into your rival. He’ll offer to take you straight to the Route 9 entrance at Cerulean City. You’ll need to go there to reach Rock Tunnel, so you might as well accept his offer. Note that declining your rival means you’ll need to make your way back to Cerulean City.
So you’ll need to go backwards through Diglett’s Cave, Vermilion City, Route 6, the Underground Path and Route 5. That’s why we suggested skipping the house, so you can grab the Leaf Stone without wasting time. If you accidentally entered the researcher’s house first, we’ll let you decide if the Leaf Stone is worth it (you can buy another later).
Once you’re back at Cerulean City, exit via the east. There will be a skinny tree blocking the way, but you can now bypass it using Chop Down.