Radio System
The radio system consists of the transmitter, receiver, on-board battery and servos. Start out with the standard range check. Power on the transmitter and receiver, leave the antenna down and walk about 100 feet away from the plane. Have a helper tell you what surfaces are moving as you move the sticks. Your helper should let you know if any servos jitter. Make sure all surfaces move in the expected direction.
Make sure both the transmitter and receiver batteries are fully charged. If the receiver battery is low, the servos will move slowly or not at all. If the transmitter battery is low, then the signal range will be reduced and you could lose control of the plane.
Fuel-powered Planes
When fuel is involved, there are a few more parts to troubleshoot. RC airplanes can burn nitro or gasoline, depending on the engine installed. These engines work like all other internal combustion engines; they need fuel, spark and air to run. Spark comes from a glow plug on nitro engines or a spark plug on gas engines. If the engine will not start, check the plug; make sure its clean and not charred. Check the battery in your glow plug lighter; a weak battery will not light the plug and the engine will not start.
Remember to check any in-line fuel filters that are installed, if they are fouled, then no fuel will get to the engine.
Electric-powered Planes
Electric motors require batteries and electronic speed controllers (ESC) to operate in RC planes. Many ESCs will either flash an LED or generate some tones when activated. If you do not see the correct flashes or hear the expected tones, that indicates a problem. Most ESCs expect the throttle to be in the off position when armed and will fail to arm if the throttle is out of position when powered on. To fix this, power off the receiver, move the throttle to the off position and power on the receiver.
If this is a new plane, make sure that the prop is spinning in the correct direction and is installed correctly. If the prop is on backward, it will produce thrust in the correct direction but far less thrust then expected. Most props have the size on the front of them and should be readable when installed.