Yeah, flat paintings is fairly routine and straightforward. Use two identical lights, one at each side, at 45 degrees angle. This will normally cancel out any small-scale texture in the base, such as canvas.
The problem with scanners is that they use one-directional light at a fairly steep angle, bringing out every last bit of texture.
Make sure the lights are positioned far enough away, so that the light is perfectly even over the entire surface. Uneven light is very tricky to correct afterwards.
If it's thick heavily textured paint (impasto), this won't work. Then you need to use soft light from the top (light source diffused over a large area). For this you can either use a large soft box, or bounce the light off the ceiling or a white wall. Again, don't put the light too close, or you'll get noticeable light falloff from top to bottom.
And then color. You really do need one of these. Put it in a reference shot:
You won't be able to match every color patch and that's not necessary; don't fuss over that. But use the neutral patches to set gray balance and overall tone. All the patches have standardized Lab values, translatable to any RGB. The third darkest gray should correspond to sRGB 124 (or 48% in Lightroom).
Calibrate and profile your monitor so that you can trust what you see! That's a whole lengthy subject in itself, which I can't get into here. But assuming you have that right: use your eyes. If it looks right, it is right.
Reflections can be killed using cross-polarized light. That's a polarizer on the lens, and polarizing film on the light sources, rotated 90 degrees relative to each other. This removes pretty much all surface reflections (only workable with flash, or the polarizing film may melt). It will boost contrast and saturation a little more than what is strictly correct, so you'll need to dial down a bit in post. Here's an example:
Oh, one more thing: LED is everywhere now, but it has one major problem for art repro: It has very uneven spectral distribution, with spikes at some wavelengths (colors) and dips at others. Old fashioned incandescent light is better, but flash is really optimal. If you have to use LED, camera profiles are needed for critical purposes, or extensive correction in post is inevitable. Avoid LED if you can.
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