Away Missions are the bread and butter of Starfleet operations. As a Captain, you'll need to oversee Away Missions frequently and learn to utilize your Crew's skillsets strategically in order to bring each one to a favorable conclusion.
Each Away Mission will offer a number of potential challenges. Sometimes you will have no choice but to pursue a particular course of action, but often times you will be presented with a choice as to how you would like to proceed. Sometimes you will have the luxury of being able to choose based simply on the most desireable outcome, were other times you may simply not be properly-equipped to take on a particular task, and need to seek an alternative.
When you order a Crew member to pursue a particular course of action, their Skill in the relevant field will be tested. Each conflict has a threshold of success that will need to be exceeded by your Crew's Core Skill, as well as a modifier randomly determined from their Proficiency range in that Skill. If they possess a Trait relevant to that challenge, this may improve their total Skill check as well. Finally, some challenges offer the possiblity of a Rare Reard (denoted by a star on the challenge node), which can be earned by exceeding a second "Critical Success" threshold. Rare Rewards can only be claimed once.
After a challenge, the Crew who particupated will become Fatigued for one turn, recusing their Skills temporarily. If they are not called upon again for the next action, they will recover in time to fully commit to the task after that. However, if they are used in succession, their Fatigue will become more severe. Furthermore, if they failed the task, the extend of the failure will carry over and make the final resolution of this Away Mission more difficult to acheive.
Becoming adept at Away Missions will require planning. You can view the potential paths of an Away Mission before you begin, allowing you to plot out the best course of action with the Crew available to you. Make sure to check for both Skills and Traits needed, as some challenge options will not be possible without a specific Trait. You can tap a node with a padlock on it in order to see what kind of Crew you will need to access that option. Some options will also require you to have been successful in the challenge immediately prior to it (ditectly to the left of it on the Away Mission path).
Take all of this into consideration when selecting your team, and make sure you're not relying to heavily on any one Crew member, and you will have primed your team for success.