You can visit the Dream World as soon as you get your C-Gear from Fennel. The C-Gear's Game Sync allows you to tuck in a pokémon in a bed. After a while, the pokémon will start dreaming, and you will be able to go to the Pokémon Global Link web site, sign up for a trainer ID and log on, then explore the Dream World with your pokémon.
You can only visit the Dream World for an hour, after which you will be required to wake up your pokémon if you have not done so. You will not be able to visit the Dream World again until 24 hours have passed since your last visit.