The lights on the instrument panel are not useful for checking this out. The headlights are the only lights that draw enough power to be useful for testing a no start condition.
Turn the headlights on, and watch their brightness while you try to start the car. If the headlights go out or dim way down when you turn the key to start; the problem is a weak or discharged battery.
If the headlights stay bright when you turn the key to start; but the starter does not run; if your car has an automatic transmission; move the shift lever to "neutral" and try starting it there. If it starts there; move the shifter back to "park" and try it again. Sometimes the starter will not work unless the shifter is moved all the way to the stop in park. If it starts in neutral but does not start in park; the neutral safety switch is failing.
If your car has a manual transmission; make sure the clutch pedal is pressed all the way to the floor when you try to start. Sometimes a thick rug or floor mat will be enough to prevent the clutch pedal from going down far enough to trip the clutch pedal position switch. And this switch, which is on the clutch pedal arm, must be tripped in order for the power from the ignition switch to reach the starter. Here's a link to a photo of this switch: http://www.rockauto.com/catalog/moreinf
This switch can be bypassed by connecting together the 2 wires that plug into the switch. That will enable the starter to work whether the clutch pedal is up or down; so you'd need to get used to the change. But older cars all were made like that.