When you have found an egg, place it in the Hatchery and wait until it is hatched. You can always speed up these actions for a few Runes. The hatchery can take one egg at the time, but it can be expanded with Runes to up to 3 eggs at the time. 

You can place the roost with the baby Dragon anywhere on the island where it is displayed in green. You might have to remove some obstacles like trees and rocks in order to make room. If it is red and you cannot find a spot, you can click on Cancel to send the Dragon back in the Hatchery or click on Hangar to store your Dragon. Dragons can also be stored in the Hangar to make room for new Dragons. To store a Dragon, select its roost, press its info and select HANGAR. To return a stored Dragon to Berk, go to the Hangar, select OPEN, choose the Dragon, and press PLACE. In order to release unwanted Dragon Eggs or Dragons in exchange for a certain amount of Fish, select RELEASE.