I use a sheet of rubbery anti-slip under my mat. It's about an inch larger than the mat all the way around to keep every thing level. I prefer Omni InvisiGrip on my rulers, but sandpaper dots are also good. There are also a couple of lines of anti-slip rulers available.
Use a sharp blade, don't cut more layers than you're really comfortable with, be sure your blade is perpendicular to the rulers edge and that the ruler is an appropriate size for the cuts you are making.
And practice, practice, practice. If you don't have fabric or scraps for practice, grab a cotton sheet at the secondhand store and use it for cutting practice. You could probably use the practice pieces as foundations for string blocks eventually.
I'd also check out the cutting tutorials on YouTube. There are some really good ones.
Good luck and have fun!