Read the "Rules for Interpretation" in the front of the Liber Usualis. Then re-read them. Keep re-reading until you understand everything. Probably 98% of what you'll need to know is there.

You can keep a lookout on eBay for the "Progressive Music Series - Catholic Edition" these were used by teachers and students during the early 20th century period. They were put together for different grade levels and published by the Silver Burdett and Company. I'm sure there are better sources today but historically it's interesting to see a method of teaching chant to children of different age groups.

It depends on where you are in the process and what you want to achieve, but I recommend the basic "Guide to Singing Chant" found in ten pages or so at the back of "The Parish Book of Chant".

or "Idiot's Guide to Square Notes" at
https://media.musicasacra.com/pdf/squarenotes.pdf

or for a more thorough approach,
"A Gregorian Chant Master Class" by Dr. Theodore Marier, K.C.S.G.

and yes, I am partial to the Dom Mocquereau interpretation of the Solesmes style.

The only thing I would say about solfeggio is that I get tripped up trying to remeber which syllable is which, especially going backwards to lower notes. I find it so much easier to use numbers for the intervals. (I know instinctively that 7-5=2.) I know some will disagree, but it just makes one less thing to think about for me. It depends on your goal and the level of commitment of the learner, I guess.

To echo Scott, Carol, and Nisi just above here, the most effective way to the mastery of plainchant notation is solfege. Most everyone can sing a scale, Scales are one of the things people have learnt by osmosis (scales and little else) in our society. Solfege gives the scale degrees a name, even a personality. It implies intervals and becomes part of one's thinking. I have taught chant classes many times and have spent a large chunk of each lecture on solfege. People seem to enjoy the challenge.

Numbers can be used instead of solfege syllables (Fr Colunba did this), but I will always treasure and prefer our heirloom solfege.

*As for those intervals, I have been astonished a number of times to find that while most people can correctly sing a scale (almost universally a major scale and not a minor one) they quite often cannot distinguish intervals outside of the context of that scale. Demonstrating and explaining the difference between major and minor intervals does not always result in smiling, bright eyed, and 'I get it' faces, but, rather, the blank faces of people who can't distinguish and just 'don't get it'. It is difficult to accept that a majority of people in our society do not have the intervalic thought that most of us here are born with. The same goes for attempts at teaching the difference between major and minor scales. With the major scale sort of 'hard wired' into their brains and nervous systems it is difficult indeed to teach people the minor scales, much less eight modes.

The OP is asking about books for someone with no musical training. Talking of Major Scales, Minor Scales, Modes, etc. will either bore the student or scare him away. Either way, you've lost a potential singer.

Sometime I think that people who are very knowledgeable about a field tend to make things more complicated than they need to when teaching beginners. You don't start a first grader off with multivariable calculus.