There's a few more things that are important to know about how drifting works in Super Toy Cars. The first thing is that each car handles very differently. The first car you get, for instance, is hard to drift with. Some cars will drift at a wide variety of starting angles and speeds, but others are harder to work with and you may find yourself slowing down or running into walls trying to start a drift. My personal favorite cars for drifting are the Shibuya WRC4 and the Schneider Panther. Note that cars that are easier to drift tend to also slip spontaneously if you take corners too fast without drifting. Managing these cars can be difficult sometimes, but there's no beating the feeling of swinging around the corners at full speed, barely scraping by each one with your boost on and running ahead of the competition. Drifting builds your boost meter, so if you can pull it off, expect to be able to zip through the stage at ridiculous speed.
Next, remember that just because you only need to use the handbrake for a moment to get your drift started does not mean you should leave it alone. If you come up to a sharp corner too fast, don't be afraid to hold that button down and turn inward as hard as you can. You'll be able to take the corner much tighter than you could by just drifting alone.
Lastly, practice, practice, practice. It'll probably feel foreign to you at first, but give it a few hours and you'll come to understand the timings pretty well. Try out different cars and see which one feels best to you in the corners. You may even find you don't like drifting at all and prefer a car that's not built for it, and that's fine! They tend to be faster straight out in order to compensate for the reduced boost you'll build and control more like a simulation style of racing game with racing lines and all that (though obviously not close to that level of realism). You can use any car in the game in the online multiplayer without needing to buy it, and practicing your skills against human opponents is a good way to stay sharp, so I'd recommend giving the online a try. The more people in online mode, the better.