You will get gold for completing certain quests and for leveling up, but unfortunately, it isn't much (usually no more than 35 coins).
A few times a day, you can play a video advertisement from the movie screen. and you have a chance of winning gold there but again it's only 30-35 coins.
You can also purchase gold with real world money by using credit or gift cards. When you are at your farm, click plus symbol next to amout of your gold. Then chose amount of gold you wish to purchase and way in which you would like to pay.
If you plan to buy gold it's best to take advantage of the offers that pop up when you start to play as they often include other items you can buy to aid your farm or a donation to your co-op.