I need to make some artwork that might be viewed in both RGB and printed with CMYK. I know I can switch the color mode after making the artwork, however, I'm concerned with color shifting when changing from RGB to CMYK or the other way around. How do you design something that you can both print and display on screen without compromising the colors?
More or less yes to both.
The CMYK gamut is almost contained in the RGB gamut. This means that almost any CMYK colour can be reproduced as RGB, but a considerable range of RGB colours are impossible to recreate with CMYK.
You may start looking here:
Generally, is there much if any color shift when going from CMYK to RGB?
Even if a given color is within the gamuts of both CMYK and RGB color spaces, that does not necessarily mean that color shifts will not occur.
You can't think of the RGB and CMYK versions of "a color" as "equivalents." There is not just one set of CMYK values for a given set of RGB values. A simple way to think about it is this: Take the product of three numbers. Now divide that number into four factors. Obviously, there is more than just one set of four values possible.
So switching back and forth between RGB and CMYK color spaces in willy-nilly fashon is not good practice. The software has to make a translation decision out of many possibilities. It does it according to your so-called "color management" settings. Even then, the software does not keep a breadcrumb trail of all of your switches back and forth. So colors can still drift.
Design for print first. Then repurpose for web. Color is not the only reason for this.