One thing to get out of the way—if you're using gas you aren't allowed at the podium just yet. That civilized flame is hardly fire anymore. It's really the canned soundtrack of grilling, lacking personality and certainly possessing no flavor you would want coming in contact with your food.
Skip pressed-sawdust briquettes and practice with chunk charcoal, which is at least recognizable as wood. Stay away from bags vaguely labeled "hardwood" and look for those that tell you what varieties (hickory, oak) you're getting so you learn the flavors.
Get a feel for how long it takes for the charcoal to settle into a cooking bed, and how much heat it will throw off.
No gauges. Learn the art of measuring heat in the seconds your hand can remain two inches from the surface. More than three seconds—too cool. Fewer? Too hot. Take your time. This practice could easily take a full season of grilling. Or two. So what? Savor the process. You're ready for the next step when you find yourself rearranging the coals instinctively. When you linger in front of the grill, eyes glazed, moving one coal an inch to the right, another to the back.
Snag yourself some oak or hickory chips from your local hardware store, or even better from a lumberyard. Try lighting your coals and adding a few chips. Pay attention to how they burn, how much flame they produce, and most importantly to the flavor they add to the fire. Wave the smoke to you. Taste it with your nose. When you're ready, cook over the burning chips and note the difference in the food you've grilled over them.

Courtesy of Grillworks
Each kind of wood you put on top of those coals will generate flame and smoke that tastes different from the others. Prepackaged wood chips are designed to add quick flavor to charcoal or gas grills, and so many printed instructions encourage wetting the wood to produce more smoke. Don't. Add dry chips slowly, allowing them to burn rather than smolder. The flavor developed by a long cook over dry wood is subtler and truer than a dousing in a steam cloud. You're grilling, not smoking.
Practice makes perfect, and lucky for your friends and family these rehearsals, if done right, will look more like a series of sooty jam sessions than practice. And they'll taste wonderful.
Ready for the real thing? Then this is your point of departure from the equipment that has become the U.S. norm. It is a lesson in frustration (or lawsuits) to attempt a full fire on a grill that isn't designed for it—so don't try. But don't run out and buy a wood grill just yet either.
Instead go camping. Use your rekindled relationship with flame to build a killer campfire. Tell ghost stories. Bring a $10 camp grill surface and some top-notch steaks. Grill them over the firewood you drag out of the woods. Let your orchestra completely dominate the stage for once.
You say that this scene isn't a fair comparison with grilling at home; everybody knows the atmosphere, camaraderie, and warmth ... it's just so different.
It shouldn't be.
Fire is a star. Let it make one of you.