When you submit two Pokemon to the Pokemon Daycare in Solaceon Town, speak with the bald man directly outside immediately afterwards to see if your Pokemon are compatible. If they are not compatible, he will say something like "They prefer to play with other Pokemon". However, if he says anything else like "The two don't really seem to like each other that much" or "They are getting along great", it indicates your Pokemon can breed and have an egg. If you use two that are the same kind of Pokemon, for example, one female Eevee and one male Eevee, they will produce eggs quickly. After you have found out their compatibility, get on your bike, and go south to Hearthome City. As soon as you leave Route 209 and enter that gateway room to Hearthome City, you should have an egg waiting for you. Go back and talk to the bald man outside of the Daycare to get your egg. If you have the Daycare Check-up Poketech app, you will see an egg appear between your two Pokemon as soon as you enter the gateway.