I also started before the rotary revolution. You need all three things: cutter, mat, and ruler(s).
Now, I had figured out I could buy gridded desk blotter pads and draw out cutting lines and cut through multiple layers of fabric at a time by pinning the open spots. They were a perfect width for folded fabric too! But then the terrible scissors I had came to play, good scissors are every bit as necessary as the sewing machine.
Anyway, if I had to pick the quilting notion I've spent the most time with, it would definitely be the seam ripper!