I bought a new Home Depot kitchen ceiling fixture but am not sure of the proper way to install it. The fixture itself has black, white, and bare copper. The ceiling wiring is red and gray(white?).
What goes to where?
The red wire is likely a switched hot -- That is, a wire connected to the hot side of the circuit, but interrupted by the switch. You should test this with a non-contact voltage tester, and have a friend flip the switch. If the wire is indeed a switched hot, the tester should ring in on one switch position. -- It should be wired to the black on the fixture.
That would make the gray/white wire neutral. -- It should be wired to the white on the fixture.
The ground wire (bare copper) may need to be screwed directly to the junction box.
Keep in mind that a ground wire is (hopefully) never used to transport current. That is, it only exists as an alternate path if something goes wrong. If your light fixture is ceiling mounted, and there is no way anyone can touch it, then it's not entirely necessary. In fact, if there is no ground to the junction box, it was installed at a time when grounds weren't required in such locations. While not optimal, it is likely safe just to omit the ground altogether.
EDIT: In reply to comments. If the originally wiring was to code at the time, then replacing the fixture does not require rewiring. (i.e. You can replace what was there as is).
But, you may need to run a ground for the ballast to work properly. If the junction box is metal, it MAY be grounded via conduit or a ground wire screwed directly to the box. If an electric meter reads 120v from red to the box itself, then the box is grounded. (Be careful, or GET AN ELECTRICIAN to check, and run the ground for you.)