"I know that black goes with black, white with white."

Not at all true all the time.

First, what was the old fixture wired to? This will be a clue.

I will all but guarantee the red is a switch leg. So the connections at the ceiling would be:
-Fixture white to ceiling white(s).
-Fixture black to ceiling red.
-ALL grounds together and to the box (if it is not already).

The red wire is PROBABLY for a fan, but don't assume anything. Black with black, white with white is USUALLY right but can change, say in switches. Again, don't assume. Testing with a meter can help. Yes the copper wires - when you say two wires, do you mean the one copper (earth/ground) in the ceiling box, plus the ground wire that comes with your fitting? Generally yes you'd put them to the ground screw on the junction box BUT - where I am, anyway, (British Columbia) you can't ground two wires to that screw, you're supposed to use a pigtail and then use a marette (connector) on all the others in the box. Pigtails are usually recommended and sometimes required in junction boxes. It sounds as if you're sort of new to this, you really can't go past a book like "wiring simplified" or the "123 of wiring" in home depot, to give you the basics. It's important that you know the basics and the techniques used, so you do stuff according to your local code.

OK good point re the switch leg. Is the fixture (I assume you're replacing one) controlled by more than one switch, for one?

Thanks for the reply's. This is a new home. Never a ceiling fixture in place. Fixture is controlled by one switch, therefore I am thinking the red wire may be a fan wire. If that is the case, do I just cap it and stuff it in the fixture recess with the other wires. Your information is appreciated.