Hmmm. The forum ate a couple of posts from last night, including one of mine. Fortunately, others have covered the major points.

Thinking about having three ranges of eight instead of 24-speeds is essential for clearing things up. After all, nobody thinks of a 4x4 as having eight or ten-speeds. We think in terms of 4-wheel high for the roads, and 4-wheel low for off-road and slippery stuff. That's two ranges of four or five gears each.

With my bike and my body and the terrain I typically ride, it works out like this: Middle ring most of the time. Small ring for steep climbs or stiff headwinds. Big ring for downhills and tailwinds. I let my body tell me which gear to use and when within those ranges, and when to move from one range to another. Your bike, terrain and body will likely tell you something different than mine do, but it will be right for you.

Beware of "double-shifting". Bad things happen when you think, "For the next gear, I need to upshift one ring and downshift two cogs" and then try to do both simultaneously. Shift one end first, let that shift complete, then shift the other end.

Slightly off-topic, a technique that helps smooth-out shifting is to "soft-pedal" your shifts. Ease off a little bit on how hard you're pedaling for the half-revolution it takes for the shift. This is because bikes don't have a clutch. Soft-pedaling effectively provides you with one, which makes all shifts smoother.