Usually it's a wiring problem involving the start and run circuits, all the way to the ballast and ECU, all the way to the coil.

Determine if you have 2 or 4 pin ballast, pull a wiring diagram from online and ensure it's wired as it should be.
Look for hack temporary repairs (wires twisted and taped together and so on).

Assuming a 727, pull the yellow wire on the starter relay to the neutral safety switch so the starter won't crank, turn the key to "start" and check coil + voltage.
Then check battery voltage across the two terminals. The two numbers should be basically the same. If the voltage at the coil is
non existant or substantially lower, there's definitely a wiring problem, bad connection, failed switch but wiring/connection problems are most prevalent.
Use a digital multimeter. If you don't own one, definitely buy one, it will help you a bunch.