You can travel with both of them together, to get them used to each other, or you can leave them together in the daycare in Solaceon Town.
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Take Eevee to the daycare center with another Pokemon. And make sure that these Pokemon are compatible with each other. Such as Eevee/Ditto, or Eevee/Eevee. You should also try breeding Eevee with other Pokemon to see if there compatible or not. If they 'get along or don't like each other much' you get an egg. If they play with more Pokemon then each other try another Pokemon because you won't be getting another Pokemon.
If leave Pokemon in the daycare, they level up according to how much you travel. More travel = more experience. Also, if you leave two Pokemon (one boy and one girl or one undetermined) and they like each other, you'll find an egg.
Yes, yes they are more than just friends. In fact Ash and Serena are the One True Pairing in the Pokemon Anime, plus the 17th Pokemon Movie as well. Oh and there shipping is called Amourshipping and is totally a confirmed/canon. Plus Ash and Serena togetherness is also confirmed/canon.
Walking, running, surfing and biking all grant 1 experience point to each Pokemon in the day care for each tile you travel over. Because biking travels over more tiles more quickly than any other form of transportation, it will level Pokemon in the day care more quickly than the other forms of transportation.
The moon stone like other Pokemon games are used for evolving pokemon such as nidoran, nidorino, jigglypuff, Evee, ect some more.