I can adjust the color of my monitor via the monitor’s own adjustment controls or via the graphics card’s software, in my case, Radeon’s Catalyst Control Center. Is there any consensus on which is the better option?
Not really, they should both adjust the colours displayed. If you are calibrating a monitor, then you might want to use the monitor’s own settings so that the calibrations stays in place even if you use it with a different computer (although most pro-calibration programs and tools will automatically re-calibrate and stuff)