Lavender Town is just a stop on your journey for now. Heal up from your trek through Rock Tunnel and head west to Route 8 when you're ready for the next city.
Pokémon nicknames
Ever wanted to go back and change one of your Pokémon's nicknames? Well, here's the only place where you can do it. Enter the building close to the south entrance and talk to the Pokémon "name rater" inside. He will only let you change names of Pokémon you captured yourself.
Like previous Pokemon versions, you cannot change the names of Pokemon received in trades.
Mr. Fuji's house
There isn't much you can do when you first walk into this house. You'll be told that Mr. Fuji is missing, but that's about it.
Pokémon Tower
Don't enter Pokémon Tower just yet. You don't have the necessary tools to identify and catch wild Ghost Pokémon yet, so you can't finish the job. Return here later after you've explored Celadon some and beaten your rival. You will find some psychic trainers under a mysterious spell and its a good place to gather some EXP. You won't be able to access the last floor, as there's an unknown ghost blocking it. You'll need the Silph scope to get past it. The Silph Scope can be found in Celadon City, which can be reached via Routes 8 and 7.
Extra experience
If you want to level up your team, there will be some fishermen with high level 20s Pokemon to the south. You won't be able to go all the way south as one of the two Snorlax in the game is blocking the path.