In the game Clash of Clans, you are awarded achievements as certain goals are accomplished in the game. These achievements include things such as collecting gold, winning battles, and upgrading buildings. You can earn rewards such as gems as you complete the achievements. The more difficult achievements will allow you to earn more gems. You can always check on your current progress towards accomplishing the achievements whenever you open the achievements screen. You want to do your best to earn these achievements quickly, so prioritize your playtime. There are three levels for each achievement, and the rewards you receive increase with each level. By simply completing the achievements available, you can earn more than 8500 gems.
Battling with other players can get you some of the most valuable achievements you can earn. There are thousands of gems available that you can earn through competing with the other players. This is by far the best way to add to your gem collection.
There are some ways to earn gems through achievements other than through battles, though. You will not earn near as many gems, not near as many as you earn through combat, but they are still worth mentioning. These achievements are earned as you upgrade your town by accomplishing certain achievements such as stealing Gold, upgrading the town hall, removing obstacles, unlocking units like the Dragon and the Archer, anther campaign completion. Each of these achievements will give you up to 20 gems.
Once you complete an achievement, be sure to collect your achievement rewards. Once the achievement is completed, a claim reward button will show up in your list of achievements. To receive your gem reward for the achievement, tap the button. The rewards need to be manually collected as the achievements are completed. Otherwise, the rewards will simply sit there. Though there is no time limit where you must collect your rewards, there is also no benefit to let them simply sit there. Check your list often to see if you have any achievements that need to be claimed.