All the tips so far have been very good. A quilt teacher told me one day to never use the lines on the cutting mat as a cutting guide. Solution, I use 2 rulers butted up together if I need them to get a wider cut.
Second thing was - always place the ruler on top of the piece of fabric you are cutting so that the good part stays under the ruler while cutting. The second tip has made all the difference in the world for me. It has almost totally stopped the slippage and bad cuts. I, also, cut, then walk my fingers, then cut again if the need arises.
I am right handed. I square my fabric up with the bulk of it to my left. Remember, ruler on the good piece. Then I either go to the other side of the table or turn the fabric so that the bulk of it is to my right so that my ruler is on the good fabric and the bulk of the fabric is to my right to make piece cuts. That way any bad cuts go into the left over fabric and you just have to trim off the edge of the piece you need. FYI...it's a lot easier than it sounds.