Should I step on the brake while changing gear?
You can enhance the smoothness of gear changes by gently easing off the throttle. There is no need to lift off the throttle abruptly or apply your brake whenever you want a gear change to take place.
These days, automatic transmissions are very smooth with upshifts even when you are accelerating hard or if you are in manual mode and shifting gears yourself.
This also applies to double-clutch-transmissions. If you are in manual mode, easing off the throttle just before shifting up will result in a smoother gear change.
With manual gearboxes, press the clutch in a fraction of a second before releasing the throttle and then go through the motion of selecting the gear. This way, you will not feel the jerk from a sudden cut in power.
The only time you should depress the brake pedal is when you shift from, say, Park to Drive when stationary. Or from Drive to Reverse. This is more for safety than for mechanical reasons.