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.
There's nothing to do here except catch a Jigglypuff in Gold and Silver (or a Clefairy in Crystal) and battle a few easy Junior Trainers. You'll note that the Pokémon Center near the entrance to Template:~ has closed, but it's such an easy area now that you won't miss it.
Not much has changed with Jigglypuff itself (although many of its old moves have been powered up a bit in Gold/Silver/Crystal). But like Pikachu, it can now lay an egg containing a new Pokémon, Igglybuff.
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.