The easiest and quickest way to teach a mouse to run through a maze is to let the mouse go hungry for a few days. When the mouse is placed in the maze, and pungent food is placed at the other end, the mouse will learn the maze as quickly as it can in order to satiate its hunger. Pet mice have an easier life than lab mice, and many pet owners do not relish the idea of depriving of them food. There are humane ways to teach mice to run through mazes, but they require time and patience.
Allow your mouse to play in the maze. It will discover and learn the maze over time. Do this every day for at least a couple of weeks.
Leave a trail of sunflower seeds along the route you want the mouse to learn. This adds interest and incentive for the mouse. Encourage your pet to follow the trail of sunflower seeds.
Place a treat at the end of the maze for the mouse. This way, it will expect and look forward to the treat.
Skip the last feeding before you want the mouse to perform by running through the maze, or simply do not feed the mouse on the day of the maze run. The combination of hunger and prior knowledge of the maze will most likely help the mouse run through the maze quickly and successfully.