Here is how I work with paintings for print:
Calibrate your monitor and then compare your calibrated monitor to (professional) test proofs so that you can be sure that you are not wasting your time.
Use Remote control software:
I have a camera which can be operated from a computer remotely via USB. The software provides a histogram and full-size preview and allows me to name the files as I take them, and save them to a network or local disk. The histogram is especially useful as you can be sure you aren't clipping bright areas (overexposure). In most cases it is better to choose an underexposed image because there is still detail to work with. Overexposed areas have no detail to salvage.
Polarizing filters on lens and lights:
Most paintings have glare. Polarization removes this but mess with red and yellow saturation.
I have a consistent photo setup so every now and again, I manually develop the best basic exposure curve and store it as a preset which I automatically apply to all shots I make in that setup. I can do this because the camera setup and lighting is all consistent and the only variable is the painting being photographed.
Archive the RAW:
RAW can be a thought of as a negative. People rarely exhibit negatives
If you cannot work with RAW: take a picture of a photo-neutral white object in the lighting you are working with and then use that photo as the "custom white-point" in your camera. Set the file type to TIFF or uncompressed (not jpeg). (if possible, check your manual for both).
Open a working copy with the curve (mentioned above) applied. Convert to CMYK. Save as lossless TIFF.
Use "select color range" tool to select the grey most neutral (in the painting, not the photo) and closest to around "75% grey value" (dark). The color range set to maximum fuzzyness. This is really the only time the Kodak card is useful. I have access to file drawers full of 4x5 transparencies with cards, and about 0% of the time is sampling or setting a magic point on them effective.
While still selected, hide selection marquee so it does not interfere with your perception. If needed, tweak levels.
While still selected, hide selection. Adjustment: Selective Color. Choose "Neutrals", (adjust neutrals ONLY--don't mess with anything else at this time) make minor adjustments. (Note that it if you think "too much yellow," you might want to add cyan and magenta rather than remove yellow.)
If you still think it needs overall adjustment, repeat the previous step but select a light non-white area and adjust "whites" instead of neutrals.
Don't make sharpness adjustments until after color correction.
There are many pigments which simply cannot be seen by a camera. Certain blues will look purple. You will need to select these areas using the same technique above and make specific adjustments on a case-by-case basis.