The goal of our mask is to stop droplets. So you want to go for something that filters those out. Then again, you need to be able to breath through your mask, so we want fabric that breathes, without letting droplets through.
In addition, we want the inside to be comfortable on our skin. So here I’d suggest a tightly woven cotton or viscose/rayon. For the outter fabric, cotton again, or a tightly woven wool.
Note that fabric that is breathable is not the same as fabric your can breath through. There’s many so-called waterproof breathable fabrics — or MBFs — that are used for outdoor gear because they repel water and are breathable but would make a poor choice for a face mask. Look no further than your umbrella for an example. Chances are it’s more from a fabric you can breath through, but only with substantial effort, which would not work for our face mask.