You need to complete the main game (beating the Elite 4) before you can access the area with the Day Care (Route 3). However you also get given a Happiny egg in Nacrene City (after the E4)
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you cant but when the egg hatches then you can trade hope this helped.
You walk, run or Bike with the egg on your team.
Well on Pokemon black and white there is a mystery egg event.
Train with the Lucky Egg (1.5x EXP) or Trade to another game with the Lucky Egg.
The Pokemon hatched is LARVESTA, the bug/fire Pokemon.