My method is to stomp the brakes, get it out of 5th and into N, pop the clutch out while rolling the right foot onto the gas to zing it, then clutch in and straight into third, and more zinging as I ease the clutch out. At certain tracks like Lime Rock, it's all about trailbraking and being smooth as you roll back on the throttle. Can't do a fast lap there any other way that i've figured out.
I don't have any grinding, screeching tires or spinning. I also pause in my shifts a millisecond, as a rule, to be nice to my transmissions.

Learning proper heel toe and double clutching is useful.
In a sense we are lucky that our cars don't brake like F1 cars and we have plenty of time to get all the shifting done.