Route 11 features several large patches of tall grass that separate Vermilion City from Route 12. To the northeast is Lavender Town, but a sleeping Snorlax blocks the east end of the route, so you'll need to find another way to get there. Inside the gate, one of Oak's aides will reward you for catching at least 30 different species with the Itemfinder, which detects any hidden items nearby. Not far from the aide is a kid who is hoping to trade his Dugtrio for a Lickitung.
A colored background means that the Pokémon can be found in this location in the specified game. A white background with a colored letter means that the Pokémon cannot be found here.
The east side of Route 2 was inaccessible when you first passed through here. Inside the house is a guy looking to swap a Clefairy for his Mr. Mime, a rare Psychic-type Pokémon. Chop down the tree to reach the gate, where another one of Oak's aides is waiting. This assistant will reward you for catching 10 different species with HM05 (Flash). Further south are two items, a HP Up and a Moon Stone.
Now that you have HM01, you can cut down the tree in the northeast part of the city to enter the museum's laboratory. Talk to the scientist standing next to the pedestal to receive the Old Amber. Take this to Cinnabar Island to regenerate an ancient Aerodactyl from the DNA trapped inside.
To the south is a convenient Pokémon Center, the last outpost of civilization before venturing into Rock Tunnel. This dark, winding tunnel emerges on the south side of Route 10, just outside of Lavender Town.