You can find it in the safari game or if you use surf and swim around in the water but it is not that usual to find it swimming.
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You can get azurill in pokemon diamond and pearl by either breeding azumarill (or marill) or capturing it in the safari zone. :]
go to the marsh and just run in mud or surf in water
After you beat the Pokemon league, go to the Pokemon mansion and keep talking to the man. After a week azurill will appear. :D
The moon stone is used in Pokemon Diamond to evolve certain Pokemon. You can find one by looking in the Underground.
You can find Aron only in Diamond.