Although they can be purchased at the store there are several ways you can get gems without putting your hand in your pocket and buying them with real money.
Achievements
Completing Achievements is one of the easiest ways to get free Gems, you will be able to see which ones you can work on unlocking at the main part of town.
Gem Pigs
When you reach level 15 you will acquire the ability to find Gem Pigs. When Gem Pigs are found and killed they will drop several Gems for you to pick up and collect.
Gem Perk
When you reach level 17 you will unlock the Gem perk which will enable you to find Gems that are scattered throughout the main town area. This means you will be able to pick up a few free Gems whenever you visit the town.
Inventory
As you progress through the game defeating enemies and collecting birds you will add new entries to your inventory which you can then activate during gameplay to get some free Gems.