Cleaning an airbrush kit according to the manufacturer's instructions is the best way. A little warm water and a brush will usually do the job. You should clean the airbrush every time after using it by filling a fluid cup with water and pressing the trigger until all the paint has been pushed out. Then you can use a brush to push the paint through if it's clogged. Once it's cleaned out, you can clean the exterior with water.  The instruction guide will usually specify what type of cleaning fluid you should use if the airbrush needs to be cleaned. The kit may include cleaning fluid that's recommended for cleaning the airbrush.