I have been using a rotary cutter and mat for 15-20 years without any training. I recently took my first machine piecing class, and one of the first things we learned was to turn over the mat so the grids can't be seen, so it's taken some getting used to, but that's how I've been doing it for the last few months.

all the time...that`s how i learned it

I was taught to use the lines on the mat and ruler, but I am more accurate when I only use the lines on the ruler. I think "the right way" is whatever way gets you the best result.

I only use the mat lines when I cut large pieces like 18" x 28"

I know it is probably slightly less accurate, but I do use the lines on my mat as well as the ones on my ruler.

I do if I have the rulers from the same manufacturer as the mat. I have noticed that if I do, the cuts are more accurate. I have some rulers from a different manufacturer than the mat, so I am careful to use only the 'correct' rulers when I'm cutting things that are critical to get exact.

I always line my ruler with my mat lines, top, bottom and side.

I line up my fabric to be straight along the grid lines of my mat but I use the ruler to imeasure the width before I cut along it.

PS I always use the same mat and ruler for cutting out each project as they all differ somewhat.

I always cut fabric,using mat lines and ruler lines. It works for me. After cutting several strips using only ruler, sometimes the fabric is no longer straight.