I'm trying to install a new light fixture for the first time in my life. I'm not sure what to do with the grounding wires, especially since the old fixture didn't have one. The directions Do I connect the 2 grounding wires (one from the fixture,which is bare copper, one from the box) together? Or, do I connect the box grounding wire to the green grounding screw on the fixture plate? Or, do I connect the fixture grounding wire to the green grounding screw on the fixture plate?
Generally speaking, all grounds can be bundled. You want the fixture, any metal boxes and conduit, and any switches all connected to the grounding conductor.
In your case, bundle or chain them all together in the simplest and most secure manner available. Don't put more than one wire under a screw, and use appropriately-sized nuts where needed.
Use short lengths of wire to make jumps from screws to nuts, or from nuts to other nuts. As element11 said in a comment,
This is what is known as pigtailing. If you have multiple grounding wires, take a short 4-6" piece of ground wire and connect it to the screw terminal in the box. Then use a wire nut to connect this wire to all other grounds that enter the box, such as the fixture ground and the ground wire going back to the panel.