If you want to improve your efficiency and endurance on your bike there are several strategies that you can use to accomplish that.
The "Spinning is winning" concept is only partly about when and how you change gears for a given situation. It is about making sure you use the mechanical advantage of your bike's drivetrain to ensure that every bit of energy you burn goes into making you move forward, faster.
For most cyclists, 90 RPMs cadence is an efficient balance point between speed and energy loss. And it can be remarkably hard to maintain that cadence in all situations. The "Spinning is winning" concept pushes that even farther. 100 to 110 RPMs is common for those cyclists following that track, and Lance Armstrong was well known for staying in 120 to 130 range.
In order to improve your metrics in this area, you need a way to measure them. If you don't have one, get a good wireless cadence capable cyclometer. Preferably one with training software included. I usually recommend the Garmin products, but they aren't cheap.
Then ride your bike. Try to maintain a cadence of 90 RPMs, regardless of your riding situation, once you're up to speed. That means when downhill, on flats, or uphill, your legs should always be moving at about the same rate.
This can only be accomplished by shifting your gears efficiently. As we move into the base of a hill, there is a natural tendency to set the body to engage with more power, to slow the cadence, and grind it out. Instead of doing this, increase your cadence slightly to build momentum for the hill, and as your cadence drops 10 RPMs below your goal (approximately) shift to a lower torque gear, and it should rise back up to your goal cadence, or slightly higher.
If you always pattern your shifts to keep your cadence consistent, you will find better aerobic fitness, and better efficiency on the bike, come naturally.
And once you can do it easily at 90 RPMs, move up to 100 as a goal, then 110. It will work.