Cavalier won't turn over, good battery

Hello! I have a 1997 Chevy Cavalier Z24, and my engine won't turn over. I drove it to school yesterday morning and it was running fine, but I tried to start it two hours later and nothing... Sometimes when I turn the key I can sort of hear the pump and everything get ready, but when i put the key to ignition everything quits. Now I know its not the battery because I got a new one at the beginning of the year, and sometimes the lights will come on (sometimes they don't, its very fickle). I tried jump starting it twice but no avail. Also sometimes when i take the key out there is this loud buzzing under dash on the passenger side. The buzzing also came on when I tried to use other electronics in the car like the door lock/unlock switch, and when i opened the glove compartment (there is a light in the glove compartment and it did not come on). So by process of elimination I found it was the door lock relay. Is that the problem? Thank you for taking the time to read this!

Bad cable connections at the battery or ground to frame/engine, this assumes the battery is charged.

Buzzing relays tell the tale.....means you have about 7 volts system voltage- so double check battery is charged- then clean and tighten battery connections as mentioned above.

Put a volt meter to battery first..should read about 12.65 volts fully charged.

I would clean the terminals... Seems like a bad ground.
The pump you hear is the fuel pump priming your system so that is normal that it comes on for a few seconds and quits

Alright, thanks guys I will give that a whirl!

Good luck... Could also be the coil, but from the rest of what you said it sounds more like bad ground.