If the battery is weak, or the battery cable clamps are either corroded or not tight on the posts, that could also cause this problem. So, before you tear into the electrical system, I recommend that you thoroughly clean the cable clamps and then run a load test on the battery. The battery should be able to supply at least 180 amps continuously for 15 seconds; with the voltage staying above 9.5 volts the whole time. If it fails that test; make sure the voltage at the battery terminals, with the engine stopped, is at least 12.4 volts. If it is below that voltage; replace the battery. If the battery passes that test; check that the voltage at the terminals, when the engine is idling, is at least 13.5 volts. If it is lower than that, and the fan belt is not slipping; then the alternator is not fully charging the battery, so the alternator should be replaced.
If it passes all the above tests, the problem is either the ignition switch or the neutral safety switch. The neutral safety switch has four wires attached to it (two of those wires are for the back-up lights). If you short the pair for the starter together, it will bypass the switch. If the shift lever is located in a console, the wiring for the neutral safety switch will be located in the console. If the shift lever is in the steering column, the wiring should come out with a bundle of other wires near the bottom of the column, and terminate in a harness plug. But the neutral safety switch and the ignition switch are both located inside the steering column, and the wiring between the two switches will also be inside that column; so it is impractical to try to bypass the neutral safety switch if it is in the steering column, unless you want to do some fancy tracing and jumpering. However, my memory is that it used to be mounted on the upper edge of the column, down near the pedals. So it is easy to replace without disassembling the column, and it only costs about $15 from Rock Auto. The ignition switch electrical portion is also inexpensive, and can be replaced without having to change the lock cylinder.