Shifting a manual transmission without using the clutch
The nice thing about owning a manual transmission car is that you can try out certain interesting techniques which are only possible on such cars because the gearbox mechanism is under your control.
One such technique is clutchless shifting, which simply means that you shift the gearbox without using the clutch to disengage the transmission. It is definitely a risky technique because it can damage the gearbox, while adding little to no value to your driving, at least for regular cars. However, it can be a fun challenge for the driver to see if shifts can be executed smoothly, and without any grinding.

My guide below only covers a shift up from 3rd into 4th, because I consider it the easiest type of clutchless shift (mainly because of the gear ratios and the shifter position), so if you would like to try the technique, try this particular one.

The idea behind it is that you (the driver) perform the shift into the higher (or next, in the general case) gear at the right time. This right time is when the current RPMs shown by your tachometer match the RPMs needed to maintain a smooth speed driving in the new gear.

Although this depends on engine and gearbox, I will assume for this guide that the tachometer shows 2700RPM in 3rd, and that you will need to be at 2000RPM in 4th in order to keep things smooth. The speed of the vehicle will not change significantly during the shift. Before attempting, you will have to identify these sweet RPM spots yourself. If you are very familiar with your car, you can pretty much ignore the tachometer and do everything by ear.

Find a nice stretch of straight road which is clear and try this:

Drive in 3rd gear and at 2700RPM. Hold your car straight and steady.
Start easing off the gas pedal and pull the gear shifter out of 3rd. You may require additional force compared to when the clutch is used.
Ease off (or even let off completely if needed) the gas pedal, to reach the target of 2000RPM.
Push the gear shifter into 4th. You may require additional force compared to when the clutch is used. Do this sharply and firmly.

After practicing, you can try to downshift from 4th to 3rd without the clutch. Obviously, you will have to tap the gas pedal to raise the RPMs, rather than wait for a drop in RPMs.

Have fun and enjoy!