The easiest way to earn additional gems is to remove obstacles from your bases. Stone removal costs gold, greenery removal consumes elixir. Once a stone is removed, it remains gone. Greenery will grow back every eight hours if you leave a 3x3 area clear inside your base. You won’t get gems each time you remove an obstacle, but you will most of the time. On rare occasions, you may find a gem box that is removed like an obstacle and always yields a tidy sum of gems.
Achievements are another reliable source of gems. You earn achievements for completing various tasks or repeating certain tasks a number of times. Scroll through the list of achievements from time to time to see if you are close to meeting the requirements for one and how many gems it is worth.