I've visited every town and city and searched all over this site and I can't find anything.
The Day Care is on Route 3. Same procedure as usual; put two compatible Pokemon in, and the female's species will determine what Pokemon hatches from the Egg (unless Ditto is being used).
You can get to Route 3 by going north from Striaton City, which means you may only access the Day Care postgame. So if you have not completed the game yet, you'll have to wait unfortunately.