In the modern world, people and goods are transported across distances by a wide variety of power systems, from internal combustion, to electrical, to rocketry, and animal (including human) power.
The first non-human powered transport was domesticated animal power, of course, reaching back into pre-history. Potential energy machines, such as spring powered engines simply store human power (when you wind up the spring), so they don't change the timeline at all.
No, to find the first example of a non-animal powered transportation system, we have to look for the first such system that operated under its own power.
If you count systems where all of the energy is expended at the beginning of the journey, then we can include the development of firearms. But if we restrict the field to systems where motive power is applied throughout the journey, then the Chinese experimentation with rocketry in the 13th century is where we have to begin. If we further restrict the field to vehicles that aren't single-use (I don't believe the Chinese ever flew a refuel-able rocket), then we will arrive at steam powered vehicles. And the first one ever was a toy built for the Chinese emperor in 1672. Once again, China figures into the history of transport.