Open a grayscale test image full-screen and look for color casts. Take your time and let the eye adapt! It could be that the ambient light is playing tricks on you, or that you're simply used to a previous color cast.
Next make a black to white gradent. It should be smooth all the way and ideally you should be able to make out all the 256 levels as individual steps. Look for any sudden transistions or color banding.
What kind of calibrator, what monitor? What are your target settings (gamma, white point temperature, white point luminance, black point)?
Luminance is a tricky one. In a "normally lit" environment you should aim for 100 - 120 cd/m², whereas most LCDs natively are up in the 300-400 range, IOW far too bright. But this depends on the ambient light. You should adjust brightness in the monitor's OSD controls, but for this the calibrator should have a way to read the value off the screen. If it doesn't, then go for a setting that gives a good match to printed output (where the white point luminance is fixed: the paper).
