Brainstorm---with other group members, if possible---the most important aims of the group. If possible, put that purpose into as many single words as possible: progress, cleanup, neighbors, compassion, environment, friendship, health, recreation. Your list of words will serve as a stimulus to creating pictures that represent your mission.
Draw sketches to represent your ideas. For example, your church youth group is conducting a park and playground cleanup, and you want an emblem for work-crew T-shirts. Perhaps a line of tools across your shirt says what you want, with your church's name underneath. A cluster of rakes, shovels and brooms inside a big heart could convey the spirit of your group and its aim.
Make your sketches as simple as possible. A good emblem is easily recognized by its overall shape, at a quick glance. That, for example, is how the triangle or circle of arrows that marks recycling bins works; you see the shape without having to look closely at small details.
Consider how well your emblem will convey meaning when reproduced in different sizes. A large established group may want an emblem that can be used on stationery, posters, shirts, business cards and even vehicles. Your emblem needs to avoid both little-blob and big-blah issues, retaining its visual impact in different sizes.
Avoid confusion with major existing emblems, both to distinguish your organization and to prevent copyright infringement. The Red Cross, Girl Scout cloverleaf, recycling triangle, stop sign and other emblems that immediately come to mind are the property of creators. If your organization's emblem reminds you strongly of something else, you're probably right---check out that nagging feeling to make sure you are not closely copying an existing emblem.