When trying to start the vehicle, it would turn over, not fire and the starter continued to turn even after shutting off the ignition switch. What should I look for to correct this issue?

Explain fire ? Ignition spark , engine turning over when key is moved to the start position . It depends on the make vehicle . Back in the old days voltage from the ignition switch when placed in the start position would go through the park neutral switch to the S terminal on the starter solenoid . Causing the starter to spin the engine over . Then the ignition system would have the job of firing the spark plugs causing the air fuel mixture to ignite causing the engine to start an run . Do you have a no- crank no start condition . Sometimes I don't know what you rookies mean , I have been fixing vehicles for over thirty years an you rookies don't talk the same automotive language automotive Technicians talk.

Does your vehicle have a remote start? Its possible that the remote start is malfunctioning. I had a vehicle with similar issues, and I didn't even know it had a remote start. (The previous owner had one, but had never told me about it, or given me the remote for it.) If you have a remote start, it should be located under the dash, next to the steering column, or firewall, or under the drivers seat. Disconnect the wiring harness from the remote start if you have one.
If you are sure you have no remote start, or if the problem continues after disconnecting it, then follow these steps.
1. Locate the small diameter wire going to the starter solenoid. The starter solenoid will be on the starter itself. This wire will be much smaller than the large wire that comes directly from the battery. This wire is is likely fastened with a round crimp connector and a philips head screw. Or it may be a clip on plug.DISCONNECT THIS WIRE. If the "clacking problem" continues with the key off, when the wire is disconnected, then you can be sure that the starter has a problem. If the problem stops (Of course it won't start either) Then proceed to step 2.
2. At this point you may want to take it to a professional who has the wiring diagrams to properly troubleshoot. But if you are up to it, Hook the disconnected wire back up to the starter, and locate the wiring harness coming from the key switch and disconnect it. If the "clacking" stops, replace the key switch. If it doesn't, then you have a short, or wire harness problem. At this point you probably need a wiring diagram and professional troubleshooting experience to proceed further.

THERE IS A CHIP IN THE KEY I BELIEVE. THERE IS A RELEARN PROCEDURE YOU CAN TRY IF WHEN YOU TRY AND START IT THE SECURITY LIGHT FLASHES. HOOK UP A BATTERY CHARGER. TURN THE KEY ON AND TRY AND START THE CAR AND RELEASE . DO NOT TURN KEY OFF LEAVE IT IN THE RUN POSITION. AFTER 10 MINS THE LIGHT WILL STOP FLASHING. TURN KEY OFF. TRY AND RESTART CAR AND AGAIN DO NOT TURN KEY OFF LEAVE IN THE RUN POSITION. LET SET 10 MINS AGAIN. LIGHT WILL STOP FLASHING AFTER THAT 10 MINS. DO IT AGAIN FOR THE 3RD TIME. SHUT KEY OFF LET SET 5 SECONDS AND START THE VEHICLE. HOPE THIS HELPS

Check starter solinoid

Sounds like your key is no longer actuating the starter/not matched to the vehicle key code, if you can turn it on but no cranking action is done when trying to start.

Depending on what vehicle you have, verify that your starter is getting 12 volts when the key is turned to Start. If no voltage is present, you may have a bad Starter Relay, Starter Solenoid, Ignition Switch, or Clutch Safety Switch.

1. Check that your starter is getting 12 volts when the key is turned to Start.
2. Check the Starter Relay to see if it clicks.
3. Check the Ignition Switch for Continuity.
4. Check the Clutch Safety Switch for Continuity.

I’m happy to help further over the phone at https://www.6ya.com/expert/jeremy_69f3cc28d95bf514

In older cars, the natural wear and tear on a vehicle can lead to ignition switch failure. If the switch is failing, you will probably not hear any clicks from the main relay, which fires the ignition. Check the behavior of the dashboard lights if your vehicle stalls. Alternatively, when the key is first turned to position two, the oil light, brake light, charge light and other warning lights will not function. You can try a test to check the integrity of the ignition switch by turning the ignition key to the 'start' position. As soon as it tries to start let go of the key. Allow it to snap back to the 'run' position and note warning lights. If they go out as the switch snaps back then the switch is faulty. if faulty it will not let power through to the starter to start car. check all fuses first. but sounds like switch.

Sounds like the ignition switch is hung in the start position, that would keep the starter spinning even after you release the switch. I would consider replacing the ignition switch. Look over on youtube for a how-to video. On a vehicle that old most likely not too difficult.

it sounds like either your having a short at the starter or the ignition switch is bad, I am leaning towards the issue being the starter, usually when the ignition switch is going bad there will be no click,

Today 11-12-08 my 2003 H2 Hummer would not start. Checked battery it ischarged. Cables tight. Tried to start with portable booster pak. No go. Tridto jump start with my wife's Suburban. Still no go.
When I turn the switch I hear a single click and that is it. No engine turning over. While trying to start with key the headlights go out. Out of ideas, Will take to Chevy garage in am. Will post solution.