Photo paper does not help if you print with CMYK inks (=standard in low cost home and office printers) Your screen image can have plenty of unprintable colors, the loss of bright blue is the easiest to notice. That's because CMYK inks are far from ideal theoretical color filters.
The printer driver does its best to keep the printed image recognizable and you see it as reduced colorfulness. If you have poorly calibrated screen, you get additional color shifts, for ex see everything on screen as hue shifted or more saturated than on a calibrated screen.
Assuming you have a calibrated screen, you should in the beginning make the design with printable colors. Many of us use Photoshop and other premium priced Adobe's programs because they understand CMYK printing limitations, can show at least somehow useful prediction of the printing result and can compress RGB images to the printable color range when the print color profile is known. Print houses and high end printer manufacturers generally give the profiles.
Adobe also gives a bunch of common color profiles as included when one installs their software. Those profiles can be useful even when print color profile is not known. I have used a profile named Euroscale Uncoated to predict what a low cost Epson printer gives. The printer presents itself as sRGB capable device, but flattens colors as it likes. I guess this is a must for most PC users. They have Microsoft Office which knows nothing about CMYK printing.
Adobe is generous. People who do not have Adobe's software can get and use those color profiles. They should be inserted to Windows, not to any application software. The color management is a part of Windows. International Color Consortium (ICC) keeps available a wide variety of color profiles which are for standardized print processes.
GIMP knows nothing about CMYK printing. But Photoshop isn't the only one for CMYK. There exists also low cost and free CMYK printing aware programs. As free CMYK aware photo editing and painting program I want to lift up Krita. See this page of it's manual: https://docs.krita.org/en/general_concepts/colors/color_managed_workflow.html
A simple method: Check if there is some reduced saturation range which will be printed acceptably. Then flatten the used color palette to that range.
ADD in May 2022: GIMP 2.10 understands CMYK print profiles, it can show the CMYK printing result and can show a warning if something is unprintable (=out of gamut) in CMYK. This all is availabe in the real time when one edits his RGB image in GIMP. The actual conversion to CMYK must still be done outside GIMP, for ex. in Krita, if one wants to use Freeware.
