Two more considerations:
The "safety guard" is there to protect the edge, not your fingers. If you push on the cutter hard enough to do serious damage, the guard will slide back without even token resistance. (And if some models do resist, it only means the cut will be deeper when they let go.) Treat the cutter as a sharp object even when the safety is on. And never put it down without engaging the safety.
No matter how sharp your cutter is, it will try to push the fabric away from itself, which builds up to a ripple preceding the cutter and, eventually, a jag when the cutter finally flattens and rolls over the ripple. Firmly pressing something flat -- usually a ruler or a draftsman's triangle -- not too far from the cutting line will force the fabric to hold still and get cut, allowing you to take long strokes without making waves.
It *should* cut on the first pass. Sometimes you have to make touch-up cuts, but never roll back and forth as if it were a pizza cutter. (I don't use my pizza cutter, come to think of it. It's much easier to hold the pizza down with a fork and use a carbon-steel knife.)
--------------------------------
My sewing style means that I rarely cut straight lines except along drawn threads. On the rare occasions that I use a ruler, I use it to draw lines on the fabric and then cut along the lines, so I can't give you advice on cutting with a ruler.
When I cut along a drawn thread, I get out my smaller cutting mat and cut the lines by drawing the cutter toward myself while holding the back of the fabric to keep it from moving. Since I can see only half a foot or so of drawn thread at time, I have to cut in short stretches anyway.
(Or do I cut away from myself? I'll have to watch me doing it sometime. At any rate, I hold the fabric on the side the cutter is moving away from.)
Joy Beeson