I'm sorry for an omission. I posted a link to the Oriental Institute at the University of Chicago's Dictionary of Demotic Egyptian, now available online. A great many other media (The New York Times, BBC, etc.) did the same, usually trying to link Demotic to something their readers might have heard of ("one of the languages on the Rosetta stone," "Cleopatra would have known it,"), etc. I did about the same amount of explication that they did.
But then, I may have forgotten my audience. This is a blog where my three-part "Why did Aramaic survive while Coptic is only a liturgical language?" series (here, here, and here) was in response to a question from a commenter on the blog. (And after all, Demotic evolved into Coptic by adopting the Greek alphabet, with a few Demotic characters, but they are two phases of the Egyptian language.) I realize I have ill-served those of my readers whose first response may have been, "first dictionary of Demotic Egyptian, great. But how am I going to learn to actually speak Demotic if I don't have a decent grammar to use with the dictionary? What if time travelers abduct me and drop me in Ptolemaic Alexandria tomorrow?" For those of you hoping to be able to converse with Cleopatra, I was neglectful.
I may have failed you, but the Oriental Institute at U. Chicago has not. Egyptologist Janet H. Johnson, longtime Editor of the Dictionary (and whom I knew many years ago when she and Donald Whitcomb were digging the port of Quseir in Egypt), has also written Thus Wrote 'Onchsheshonqy - An Introductory Grammar of Demotic.
Once again, through the generosity of The Oriental Institute, the text is available online as a .pdf.
So now you have pretty much all you need to teach yourself Demotic. Except maybe tapes to show you how to pronounce 'Onchsheshonqy. Seriously, though, I want a copy.