Instructions
Use the computer in any Poke Center to place a Cyndaquil in your main Pokemon lineup. Check to see if the Cyndaquil is a male or female. This will be indicated by a symbol next to the Pokemon's name. Mars symbols, with an arrow pointing to the side, indicate male Pokemon. Venus symbols, with a circle at the top, indicate a female.
If the Cyndaquil is female, select a male Pokemon to breed her with. Pokemon eggs always hatch into the lowest evolutionary form of whatever Pokemon the mother is. If the Cyndaquil is male, select a Ditto and place it in your main lineup. Dittos are genderless and will breed with any Pokemon. You also can use a Ditto with a female Cyndaquil.
Walk to the Day Care Center. The Day Care Center is on the east side of Route 34 in the "Gold" and "Silver" games.
Talk to the woman inside. She will ask if you want her to raise a Pokemon for you. Select "Yes," then choose the Cyndaquil for the woman to raise.
Talk to the man inside. He will also ask if you want him to raise a Pokemon. Select "Yes," then select either the male Pokemon or the Ditto for the man to raise.
Walk out the back door of the house (the door you didn't use to get into the house). This brings you to a small courtyard.
Walk up to a Pokemon and check its status by hitting the A button. A message will appear. If it says "Pokemon A cares for Pokemon B," "Pokemon A shows interest in the other Pokemon," or "Pokemon A is friendly with the other Pokemon," the Pokemon are getting along and will likely create an egg. A message saying "Pokemon A is brimming with energy" or "Pokemon A has no interest in the other Pokemon" means that the Pokemon will likely not create an egg. If this happens, select another Pokemon to breed with the Cyndaquil. Keep trying until you find a pair that gets along.
Wait for the Pokemon to breed. This can take several days. You can check in at the Day Care Center to see if it has happened yet.