This kind of simplifying can be done by using Photoshop's Cutout filter. It needs a high quality image If you want a high quality result. Cutout is already discussed in other answers a long time ago.
You can trace the image to vector domain. Even a low-resolution photo can give printable results.
To be able to trace the image, adjust it's levels to cover a wide range of brightness walues in all important image areas such as faces and technically important details. In this low resolution photo the face really needed a contrast boost to make the smile visible. Without the boost it would need more levels to the tracing.
(BW. This image is a fully legal copy)
I added an adjustment layer "curves" for the contrast and a layer mask to prevent the rowing machine to burn full white. Photoshop also is the right place to remove the unwanted details such as the bottom shadow (=done). This all is valid also for cutout filtering.
Next do a greyshade tracing in Illustrator or other vector image editor.
I want more control, so I make a few single treshold BW tracings. I put one treshold to near white, other is quite dark and a couple of traces in the middle. The tresholds are adjusted for good shaping of the details. The face is most important here. I want to use Inkscape instead of illustrator due much more easier control just in this case. Illustrator is used, if I want trace strokes. Inkscape traces easily only fills.
Next I align the traces, make them partially transparent and give colors to them. Also a background is added. Note that the coloring of the traces can be arbitary. The path simplifying is useful. I took Inkscape's default simplifying.
Trace stacking order: Largest (=brightest treshold) is at the bottom and the smallest (=darkest treshold) is on the top.
Lousy preprocessing and treshold selection both appear as missing or unwanted parts. They can be partially still fixed by drawing patches that cover something or replace it. Here I added a floor and a shadow (=was removed in the beginning). The colors are adjusted.
The patched result:
Because this is a vector image, it can be exported in as big size as wanted.