Currently the Earth Dragon Egg can only be obtained by buying it with rubies, through special in-game promotions and events, or by collecting Earth Dragon crests and converting them into an egg.
Once you you have your Great Dragon armored and capable of attacking, you can start attacking level 6 and higher Hills in hopes of getting Earth Dragon crests as a random drop.
You must already occupy your maximum number of wilds: if your attack occupies the wild, you won't get a random drop. Each attack must be accompanied by your Great Dragon. The Great Dragon must heal before you an attack again, so this can take a long time, as you need 100 crests, and not every attack will drop a crest. There is hope, as some drops include more than one crest (I have gotten as many as six in one drop), but most of the time it is one crest or no drop.