What they said.
I am a grass man myself. My yard grows like crazy so i have plenty for mulching composting and grasscycling.
You could mix shredded paper and grass for eve better mulch, you would have your greens and browns all in one nifty package. I myself wouldn't use shredded paper without grass due the windy nature here. It would be everywhere. I add bags of it to my compost but only if i have a bag of grass clippings ready to hold it down.
Another thing if you go with grass clippings it might be best to let them sit for a short while to brown up before adding. They break down fast and when fresh will actually rob nutrients out of your soil of course they will be replaced during the decomposition process but than you are not ahead but back to where you started. I have put on fresh clipping my whole life without any disaster but from now on I am going to let the age a bit first. Just to be safe.
Oh yeah leaves too what am I thinking another good mulch added with grass even better.
Same with straw.