Here's what I do (disregard the charts at first... they are there for mathematicians and confusion)

Find a front chain ring (left shifter) that provides you with a comfortable range of gearing (likely the small one or the middle one)
Keep it there . In other words, for now, ignore the left-hand brake-gear change lever (think of it as a front-brake lever... and remember: when you are flying down hill, engage the right or BACK brake lever first or you will end up cartwheeling on your bike!)

Now, use the gray shift lever on the right to move into easier gears (up hills or coming to stops); use the smaller/black shift lever to move into harder to pedal gears (down hills or on flats or with a tail wind)
LArge-gray lever on right: EASY gears
SMall black lever on right: harder gears

I have only two chainrings in front. I rarely transition to the larger chainring... only if I am on a long flat or down hill and want to 'tweak' (increase) my candence (pedal rotation speed) a bit. I will shift into the easiest gear in back and then shift up to a the larger cog in front (the difference is dramatic!) As my conditioning gets better, I may hesitate shifting up or down to allow my legs to work a wee bit harder... but mostly I use the gears to keep my rhythm smooth and seamless... by feel

That's my approach. I'm not counting teeth or analyzing ratios or such. I am listening to my legs and cardio rate, and adjusting for my comfort and speed. I adjust the gearing based on my needs not on some perceived mathematical scheme. All that rot is interesting later, over a beer with the biker geeks... while I am on the bike, I use a more intuitive approach.