Clutchless shifting is neat and all, but the risk of damaging your transmission is not worth the tiny amount of wear you’ll save.
The real way to improve on most people’s shifting is to rev-match, which basically amounts to getting the engine spinning at the right RPM for the gear you’re going into. This means that if you’re going to downshift from 4th to 3rd, you’re going to be going from (for example) 2500 rpm in one gear to 3500 rpm in the other. There’s a right and a wrong way to do this.
Most people do it like this: put the clutch in, letting the rpm’s drop to near idle, then put it in 3rd and let the clutch out, causing the clutch plate friction to get the engine up to 3500 rpm. As this happens, the driver’s thoughts (“I am a badass race-car driver”) are drowned out by the clutch whining in agony, a sound most people seem to think means “great shift, dude”. The car slows down quite a bit in the process, since it’s basically using your clutch plate as a big expensive brake pad, which is bad for it.
The correct way to downshift is to put the clutch in, then while you’re shifting from 4th to 3rd, you give the car just enough throttle so the rpm’s get up to the right spot (3500 in my example, but it varies based on your car and your speed, and is just something you have to get a feel for), so when you let the clutch out, there is negligible wear on the clutch plate. You then let off the gas, which uses the engine compression to slow the car.
There’s this idea out there that downshifting is 1) a really effective way of slowing down, and 2) how the cool guys do it. It turns out that the incorrect method of downshifting is actually much more effective at slowing your car down (although at the expense of your clutch), so people end up with the idea that since that method of downshifting “works really well”, it must be the right way.
Not only is rev-matching a great skill because it’s good for your car, but it’s useful in high-performance driving, it makes the ride smoother for your passengers (if you’re good at it), and it’s fun.