why the ignition light and fuel cap light stay on when driving

Not sure what the "ignition light" is - unless you mean the check engine light. The fuel cap light is suggesting you check to ensure your fuel cap is tightly secure. If you have checked it and is, you have some kind of leak in your evaporative emissions system, which will cause the engine light to come on as well. This system will need to have troubleshooting performed to determine the cause of failure.