Are there any free/ open source image designing programs that you can recommend?
For a hand drawing one giving tons of control over stroke settings and growing, is MyPaint. Scribus, a layout/publishing tool, is told to be one of the very few supporting CMYK well. Indeed, some people do their design in Inkscape, and export to Scribus for the CMYK features. Believe it or not, I have used Blender from time to time as a helper in some graphic work (rarely for a design).
Was forgetting sk1, an illustration tool supposedly better suited for print with CMYK support. It might still be a work in process and maybe not in all platforms, yet (only linuxes at the time of writting this answer, next two releases will bring Windows and OSX versions). I'd keep an eye on it.
Krita is another open-source raster graphics editor. It is mostly aimed at digital painting. Among other things, it has features like powerful brush engine, symmetrical drawing mode and warping tools.
Gravit Designer is presenting itself as a cross-platform design tool for the 21st century, and I completely agree with that description.
It is a vector based program, but you can also import raster images.
It runs on different platforms.