Motors and motor controllers designed for robotic use would be a good first stop - what you are talking about is somewhat like an electric golf-trolley but scaled down. Robotics forums are probably a good place to find out about axles/wheels/motor possibilities, and here for
general design ideas.
Start out listing the requirements (speed and power, human interface, safety issues), then a rough parts list (such as battery, motor hardware, some sort of handle with switch or other sensor), then try to identify possible parts, then try to sketch a rough mechanical design, and then an electrical / electronic design.