Pokémon Red and Blue/Diglett's Cave

'With Route 11 blocked, you must take the roundabout route to Lavender Town. Diglett's Cave connects Route 11 with Route 2, emerging near Viridian Forest. Head north to Pewter City, then east to Cerulean City and Route 9. Now that you know how to Cut, be sure to visit the secret lab behind the Pewter City Museum.'

Through the cave
Diglett's Cave connects the west side of Route #11 and the north end of Route #2, near Viridian Forest. The path is easy, but and  can be annoying, so remember: Flying creatures are immune to Ground-based attacks, even Dig.

Back to Route #2
When you exit Diglett's cave, you'll be in a familiar place, the north end of Route #2 near the exit of Viridian Forest. Now that you have Cut, the remainder of Route #2 is available to you.

The building south of the cave exit has a guy who is willing to trade you a rare, in exchange for an in Red/Blue, or a  in Yellow.

Further south, there's another building in which you can get HM 05, which contains Flash, the item you came for. This is used primarily to light caves, but in battle it will lower the Accuracy of your opponents' attacks. Its current owner, another one of Professor Oak's aides will only fork it over if you have 10 or more Pokémon. Below that house are two item balls which contain an HP Up, and a rare Moon Stone.

While you're in the area, head back to Viridian City and cut down the tree in the lower left corner. Walk inside the area and talk to the man, and he'll give you TM 42 with that Dream Eater for your. Cut aside another tree in Pewter City to get the Old Amber from the Museum Curator. When that's all done, either travel through Mt. Moon, or cut back through Diglett's Cave, and head to Cerulean City.