Try turning the headlights on; and watching their brightness while trying to start the motor. If the lights stay bright when the key is turned to the start position, this means that power is not reaching the starter solenoid when the key is turned. This could be caused by a defective starter relay, or a defective clutch pedal position switch (on manual transmission cars) or a defective neutral safety switch (on automatic transmission cars). The starter relay is not expensive; and is usually the culprit; so if the lights stay bright in the test; I would first replace the starter relay. If that doesn't fix it; I would replace the clutch pedal position switch or the neutral safety switch (depending on the type of transmission your car has).
If the headlights dim way down or go out when trying to start; then the battery cables should both be disconnected and thoroughly cleaned with a tapered reamer type battery cable service tool.