Try replacing the starter relay (which costs less than $10). If that doesn't fix it; replace the neutral safety switch.
If you want to double check in advance whether the above parts are needed; have a mechanic check the wire from the ignition switch to the starter solenoid; to see if it is supplying power to the solenoid termninal when the key is turned to the "start" position. If there is no power on that small diameter wire to the solenoid when the key is turned to the start position; then it is one of the two parts listed above.
If there is 12 volts coming from the ignition switch to the solenoid when the key is turned; then the starter solenoid or the starter motor is defective.