The biggest mistake that new supervisors make is to think that they have to do it all themselves. This usually drives those employees you supervise crazy. Gain an understanding of problems through active listening. There is no way to solve a problem unless you understand it. Make sure to get input from as many people as possible within the department, because there will be as many different views as there are people.
Great managers are fair.
Be careful about preferential treatment. It tends to be a blind spot for many of us. Demand good work from everyone and don't tolerate lazy performance. There is no one to blame in that situation other than the supervisor. Support and back up employees to upper management.