There are several ways you can get more Gold coins, some easier than others.
Sell Birds
The obvious way to get more Gold coins is to sell off any birds that you don't need. The amount of Gold Coins you will recieve for the bird will depend on its rarity level which is indicated by the amount of stars it has. Every bird is worth at least 10 Gold Coins and this will increase by 10 Gold Coins every rarity level. For example, a four star bird is worth 40 Gold Coins and a five star bird 50 Gold Coins. Selling birds also allows you to make space in your inventory.
Missions
Obviously completing missions will gain you Gold Coins but once you reach level 8 and unlock the game's Hard difficulty setting the amount of Gold Coins you will earn for completing battles will multiply considerably.
Dungeons
Completing Dungeons will guarantee you earning a large amount of Gold Coins but to enter the Dungeon in the first place requires 5 birds of the same colour and a Team Power rating of at least 2500. You will aslo need at least 10 Dungeon Keys, or 15 if you want to skip the battle.
Timed Adventures
Once you have reached level 6 you will unlock 'Timed Adventures' which will enable you to earn some extra Gold Coins by sending Scouts out to complete them. Although the payout is low and they take a few hours to complete taking part in 'Timed Adventures' is still a good way to earn Gold Coins without tying up your more important birds.
Pigs
Try to hit the pigs on the sides as doing so will give you extra Gold Coins and is usually very easy to achieve.