Route 3 leads to Mt. Moon, where your Rival is waiting, ready to battle. You'll run into a handful of Trainers with Flying-, Poison- or Fire-type Pokémon, so you might want to backtrack to Pewter City before continuing on to Mt. Moon.
Route 3
There's nothing to do here except catch a Clefairy in Crystal and battle a few easy Junior Trainers. You'll note that the Pokémon Center near the entrance to Mt. Moon has closed, but it's such an easy area now that you won't miss it.
Clefairy has a couple of new moves, including level 4's Encore (which repeats your opponent's last move) and Moonlight, a healing move that is stronger at night. And now they can lay an egg containing a new Pokémon, Cleffa.