You can power a small toy car in a variety of ways, and each has benefits and drawbacks. If you are starting from scratch, research and experiment with different power sources before building your car, because the strength of the power source will determine how light your car must be for it to move any meaningful distance. Also consider whether the power source you're interested in requires replacement, or if you can use it repeatedly. Also consider the cost and availability of materials around you, and be resourceful with your design.
Use the energy stored in an inflated balloon to power a lightweight car. Mount the balloon on top of the car in such a way that the mouth of the balloon is in a rear-facing position, and can be inflated in place. Inflate, then set the car down and release it.
Use the elasticity of a rubber band to make a simple windup car. Anchor one end of the rubber band to a spot on the bottom of the car, and loop the other end over a peg attached to one of the axles. When you hold the car on the ground and pull it backwards with your hand, the rubber band will wrap around the axle again and again, building up tension as it stretches. Release the car, and the rubber band will unwind rapidly, spinning the axle and wheels and pushing the car along. After completely unwinding, the rubber band should slip off the peg, allowing the axle to continue spinning until the car runs out of momentum.
Mount a spring-loaded mouse trap on top of the car's body and use its stored energy to push the car. This is very similar to the rubber band technique. Run a string from the trap's spring-loaded jaw to a peg on the rear axle, then pry open the jaw as you move the car backwards on the ground, wrapping the string around the rear axle. When you release the car, the jaw will shut, pulling the string and spinning the rear axle, which propels the car. See the References section for an example.
Use a compressed CO2 canister for a hyped-up variation of the balloon method. The small mouth of the canister should face backward on top of the car, and the body of the canister must be firmly buttressed on all other sides to keep it in place and transfer all of the force from the expanding gas directly into the car's body. Starting these cars can be tricky, as you must pierce the end of the canister to release the CO2. If a special launch pod designed for this purpose is not available, a nail is a low-tech solution.
Use a model rocket engine in place of the CO2 canister or balloon for even more thrust. These small solid-fuel engines are affordable and are available at hobby stores. Start the car just as you would a model rocket, attaching the leads to the engine with one held in the end with a plastic plug, then igniting the fuel from a safe distance using a battery-powered igniter.
Power a DC electric motor with a battery to propel the car. Most remote-control cars are powered this way. Connecting the motor's axle to the car's axle can be tricky without gears or other special parts. A solution is to use a three-wheel design, with the wheel that's attached to the motor centred on the car's body by itself, and the other two holding the frame up. Experiment with a motor and battery before building the car's body, as the strength of such motors varies greatly, and if your car is too heavy for the motor it will not move at all. Adding batteries may produce more force from the motor, but will add weight to the car as well.