Cluster Lighting issue.
 

I have a 2000 Chevy Cavalier. It has been having a problem for quite some time. There are times where the instrument panel lights will not come on when I turn on my headlights. I then drive for a while and pull off the side of the road. Turn the headlights off. Take the keys out of the ignition, put them back in, start it and turn the headlights on and all the cluster lights come on. Now when I turn the headlights on, they do not come on at all and I get a small beep in the center of the dash as the cluster lights flicker for a brief moment and then shut off. But the headlights work just fine. Any suggestions?

First off, check all the grounds on the car. That is, disconnect them, clean to shiny bare metal, re-install.
The BCM (Body Control Module) probably controls the panel lights, along with a bunch of other stuff. It's right there next to your left foot when you drive. It's a PITA to remove, but I would remove it, unplug the 3 wiring plugs, clean it all up with contact cleaner, check for corrosion, then re-install.
Look for corrosion on the fuses underneath the little door at the far left end of the instrument panel.
If all else fails, it could be the MFS (Multi-function Switch), the turn-signal switch assy.