You would typically double-clutch in downshifting, not so much in upshifting. I assume that you shift smoothly and reasonably slowly when upshifting... i.e. out of the lower gear, then pause a short while in neutral, and then into the next gear. That is easy on the synchromesh parts. AND, you do not grab the shifter with your whole hand-grip... you only motivate the stick with two fingers, in the direction you want it to go in... Let the synchro do its job, and do its job AT IT's PACE.
For double-clutch downshifting typically you kick-in the clutch and take it out of gear, placing it in neutral. Depending on the clutch and/or transmission you may have to only kick-in the clutch half-way (to take the stick out of gear... and to place it into neutral) before you let the clutch out again. So... 1/2 kick of the clutch and place stick into neutral and then release clutch pedal fast.
Now with the clutch pedal NOT pushed, you rev up the car to more or less exactly match the revs you would do when in the lower gear, if it were in that lower gear with clutch out... It is not so much a jab of the accelerator... Rather, you push and hold the accelerator to the right point as described. That takes a bit of practice and judgement on your part. In any case, with the accelerator continually held at the correct (higher) rpm, you decisively push in the clutch fully and really shortly after (or during the clutch-pedal-stroke down) you get into your lower gear on the stick. You are STILL holding the acc. at the correct rpm. Then you let out the clutch, fairly quickly (and indeed, if your rpm is correct, you let it out really quickly)... and there is neither any acceleration of the vehicle, nor any compression-braking of the vehicle... because your rpm is perfect for the roadspeed. You can see that this means very little wear on the synchronizer for that lower gear, and it also means very little wear, actually zero, for the clutch disk, too.
Once you have perfected your double-clutching (A.K.A.double-declutching in much of the world except the USA or Canada) you can see if you want to do "heeling and toe'ing" combined with double-clutching. That is the 'real fun ballet-dancing move on the pedals...