One of the hallmarks of an advanced driver is a smooth style with manual gear changes.
A car’s gearbox is a wonderful piece of engineering, rarely casing problems during the expected lifetime of fifteen years, withstanding hundreds of thousands of shifts up and down the ratios. It’s so well developed that all you need to do is gently guide where it wants to go.
Yet "snatching" at gears is common, causing an abrupt and a less than comfortable ride for passengers. Another problem is drivers who hold the gear stick constantly, refusing to let go, in order that they can make the gear changes as quickly as possible. That hand would be more use on the steering wheel.
To make the gear change smoother, let the gearshift pause for a second or two as it crosses the "central gate" or neutral zone. This alone will make your gear changes smoother – there’s no need to rush them.
Secondly, guide the gear lever with the palm of the hand semi-open with the thumb sticking out. Guide the gear lever away from you for 1st to 2nd. The thumb should be pointing down. Now guide the lever towards you for 2nd to 3rd,3rd to 4th and 4th to 5th (or even 5th to 6th). Now the thumb is pointing upwards.
For block changing (when you "skip" intermediate gears) keep the thumb down for 4th to 2nd.
And remember, around town, you can often stay in third gear and respond to new hazards early by easing off the throttle. Adjusting your speed with a fine throttle has all sorts of advantages.
As a check to see how smooth your gear changes are, take a glance at your front seat passenger’s head. Are they constantly "nodding"? If so you need to make your gear changing smoother and less hurried. Your passengers (and transmission system!) will appreciate it.