Currently, there are only two ways to get free gems: Unlock achievements and tap on treasure chests scattered all around the map. There are various achievements to unbolt and many involve simple tasks like rescuing friends, calling buddy characters to help you, traveling 10 miles etc. Completing one achievement opens another, but it will be slightly more challenging than the previous one.
You can also earn free gems from treasure chests. These chests appear from time to time every day.