Mesh/Geometry quality tried searching but couldn't find anything.
I do not have a 3d printer so can't really experiment on my own, which means that when I am going to order a 3d print I want to get it as good as possible. So, my question:

Do quality of geometry matters when 3d printing? Will 3d printer only print quads, or ngons are fine? Are there shapes to avoid?

Cheers :)
M.
Welcome to the site!

In regards to quads, vs polygon. People will often reduce the overall detail to make it easier to print. But so long as after you export it to a STL and verify that your Manifold edges were done correctly and what you though was solid is solid, you should be good to go. As near as I can tell so long as you can export it to STL then it doesn't matter what meshing you use. That said I see Polygons more than anything.

Worst case you can run it through a STL repair program and it will make the required changes for you. Usually I used these tools to fix poorly rendered files.
My favorite is Nettfab. which is now part of microsoft.

You can verify if it will print by downloading slic3r, then "slicing" the file. After that you should be able to view a layer by later output.

Article talking about quads vs triangles from design

Shapeways article on preparing blender files for 3d printing