There are loads of people doing this, and if I lived closer to a big coffee shop I'd consider doing this, but I live out of town.
Of course, I could just ask for a bag or two of grounds every time I go there.
The great thing about coffee grounds is that they are pre-pasteurised, so you can really just inoculate and go.
This seems to be a good guide on how to go about it:
https://www.growveg.co.uk/guides/growing-gourmet-mushrooms-at-home-from-waste-coffee-grounds/
This is also a decent guide, and the author suggests
Hypsizygus ulmarius (elm oyster mushrooms) and
Agrocybe aegerita (pioppino/black poplar mushrooms):
http://madbioneer.blogspot.co.uk/2011/01/coffee-ground-mushroom-spawn.html
I'm not sure whether
Stropharia would work on coffee grounds, but by all means give it a try.