2000 Cavalier occasionally doesn't start

I have a 2000 Chevrolet Cavalier with a 2.2L SOHC 4 Cylinder. Recently, I've been having problems with starting the car. When the car hasn't been started in awhile, it usually starts up fine on the first try. However, if I have just shut the car off, and try to start it again, the engine fires and turns over, but refuses to start. I usually have to wait about 10 minutes or more before I'm able to start the car again. Any ideas on what the problem could be?

Also, I've noticed that when I turn the key forward (to the point where you can turn on your lights and see the odometer and stuff), a lot of the gauges behave oddly, turning on and off and stuff. When the key is in that position this thing (i don't know much about cars) makes a weird kind of buzzing noise.

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