Arlene and I have revised our old article that ran in Crisis, May 2006: "An Idiot's Guide to Square Notes." It was actually commissioned by the staff because they couldn't make heads or tails of the music they kept seeing in chant books.
The link remains the same but in the past, we had up a PDF from the final publication, and it was a bit bulky. That is now replaced with a new version that looks more like a manuscript but has a much faster load time. It is 12 pages.
I'm still very happy with it overall because it seems to be the most accessible source for answering the precise questions that musicians have at the very outset of neume reading. I can recall when I first started trying to read chant, the Liber Usualis tutorial was almost useless to me: so many technicalities but no clear explanation of how to figure out the melody! The same was true of many other tutorials that I ran across. It turns out that academic or monastic chant specialists are not really the best people to ask for a quick tutorial, or so it seems.
Now, if you are seeking something that is both clear and more in depth, the best source is the tutorial in the back of the PBC. But this guide will get you pretty far actually.