Boris - you can practice in your own car, and at first you don't have to brake. Try this;
Along a nice straight road, perhaps 40mph in 3rd gear. You're going to leave the brake pedal alone for the moment, just practice on the principal.
1) Now, try changing to 2nd as "normal", your right leg doing nothing. You'll notice the car jerking when you release the clutch in 2nd, this is because the gearbox winds the engine up due to the higher ratio between 2nd & 3rd. Try that a few times, listen to the engine note/pitch when you change.
2) Do the same as above, nice and slowly, however this time when you press the clutch in and change to 2nd, squeeze the accelerator pedal gently and bring the engine revs up to match where they would be in 2nd gear. You'll notice now that you're brought the engine revs up to meet 2nd gear instead of the car doing it for you.
This is called a "match rev gear change" and is kinder to the transmission/driveshafts etc.
3) An advanced way of doing this is to be braking with your right foot and using your heel to blip the accelerator. This is called "Heel & Toeing". Don't try this yet, possibly get someone to show you. And never try it on a public road untill you get proficient.
4) Just for completeness, you don't need to read this, and it generally isn't done on a road car, there is also "double de-clutching". Its similar to number 2 above and is intended to change down in vehicles without syncromeshes between the gears. The reason you can actually change gear in a car the 'normal' way is due to a syncromesh. My dad learned to drive army trucks that didn't have these syncro's. So you have to blip the clutch while changing down in order to spin up the shafts in the gearbox so that the gears meshed properly. If you can't do match-rev changes yet don't even consider this one, it isn't really used these days. However I was fortunate to learn this tecnique as I learned to drive and so all the above comes very naturally to me and I use mixtures of all these on the road and track depending on the circumstances.
Dave