There are equations used to determine the minimum number of moves based on distribution of certain colors. Algorithms. I never got into it but some mathematicians did when I was in school. Those guys would figure it out on paper and then just do it, no trial and error at all. The math necessary to do it can be done by just about anybody too but like with everything in math you can of course make it as complex as you like.
But all those guys that finish 'em in seconds are all using algorithms, basically just memorizing a series of moves and going from there. It's the same way someone calculates pi to infinity. I worked out pi's infinite series once and all you have to do to name off each decimal point is to solve for x each fraction in that series, but x is just 1, 2,....,n-1, n. It's simple once you know how.
If doing it the old fashioned way, I think you're supposed to try and get one field colored in, then work on the others until they're all solved. Can be easier said than done too.
I'm also guilty of swapping the stickers too!