As TCampbell pointed out, a grey card is the easiest and most economical means of setting the White Balance properly in the camera (take a photo where the grey card is most, if not all of the frame, then set Custom White Balance (CWB) to that image). That will get the correct colors in the JPGs in the camera. If shooting RAW, then the White Balance can be set during post processing in a similar means.
The X-Rite Color Checker or similar products will give absolutely the best results because it has an exact 'known set' of colors on their multi-color card and the software that comes with it will set the colors in the computer to match the known set of the card in the photograph you set it to.
Using either method for WB will get the colors right in your computer...I'll call this step one.
Then one may 'fiddle' with the colors based on what they see on the screen. That's where having a calibrated monitor makes the difference. If, for example, your uncalibrated monitor makes everything a tad greenish, your eye will lead you to setting the WB to correct the greenish tint. The calibrated monitor will accurately display the colors that are 'in the computer', whether they are right at that point (set by grey card or color-checker) or not (Auto White Balance in the camera, typically). That's why 'step one' must be correct AND the monitor calibrated before adjusting the colors ever-so-slightly to, perhaps to tweak the blue saturation, or whatever. Call using a calibrated monitor and minor tweaking 'step two'.
Step three is printing. Mr Campbell also noted every ink, paper, and printer combination will render colors slightly differently. So, it's necessary to properly calibrate the printer/ink/paper combination the images are to be printed on. I have a calibrated monitor and use grey cards and/or Color Checker as needed. I don't do any 'critical printing' so what my Epson printer prints is usually 'good enough' for me and those who get my amateur prints.
Professional photographs used for publication by newspapers, magazines or online services usually have their own 'rules' about how they want the colors rendered, and even what 'color space' to use (I think that's the term, I've never fooled with it, personally) They will likely insist on using a Color Checker or other similar product to guarantee accurate color representation and THEY would likely be the ones to tweak the colors per their needs and desires for publication.
Lastly...as Photoguy99 alluded to, getting the colors perfect in every shot is very difficult. This is because except under controlled studio conditions, lighting perhaps 5 feet from where the grey card/color checker was photographed my be slightly different. Using the same settings as the previous picture, for example, may result in colors 'just a little off'. Throw in the 'cycling' of florescent lighting, and two photos taken not even 1 second apart will have different colors. I have to deal with that in some of the church events I photograph. That's when a flash is handy to 'override' the ambient lighting to control the colors.
Clear as mud? Add in the effects of over/under exposure that may 'degrade' colors slightly, and noise, it can get complex.
For what it's worth, my digital 'education' started with learning to get the best exposure results with what I had. That meant getting the exposure triangle 'automatic in my head' first. I didn't even know what RAW or White Balance was at that point, and had reasonably satisfactory color results with Automatic White Balance (AWB) on my camera. I sometimes spent too much time trying to fiddle with individual colors in post processing until it looked right, without understanding WB. That was my next big step in the process...WB. Then on to RAW and working with the 'full pallet' of colors rather than what the camera produced for JPGs. In short, I took one step at a time. I think if I tried to take on everything at once, I would have put the camera down and used my cellphone.