Once you install hydraulics (any level, but 3rd is the best), you can use it for faster accelerations at a start of any race. Don't use it for free roam as it's usually faster to just do a burnout using nitrous.

Basic wheelie: Burnout bonus: 50-250

1. Hold hydro pressure and brake/reverse buttons to raise the rear of your car.
2. Once the rear is up at maximum height, release both buttons.
3. Now hold gas and once the countdown reaches "1", add nitrous and hydraulic bounce.

If you're in a free roam, you also have to hold handbrake, or just don't raise the rear and instead use bounce/nitrous/gas combo right away.

Advanced wheelie for 5 gears Corolla (easy mode): Burnout bonus: 150-1000

1. Just as the countdown starts, hold gas and hydraulic bounce.
2. The car will bounce three times and when the race is about to start, add nitrous.
3. After you get up to 60-70kph, shift to 2nd and keep holding nitrous up to 3rd.
4. If you timed it correctly, the third bounce should be synced with the race start and you should get up to +500 burnout wheelie (Usually over 300). However, the game often doesn't count the wheelie and you won't get any nitrous. If that happens, restart (if it's IL).

Advanced wheelie for 5 gears Corolla (hard mode): Burnout bonus: 150-500

1. Hold hydro pressure and gas to slightly raise the front of your car.
2. When the countdown reaches "2" (almost instantly after you raise the front), hold hydraulic bounce.
3. The car will bounce twice and when the race is about to start, add nitrous.
4. Right after you start, shift to 2nd and keep holding nitrous up to 3rd.
5. If you timed it correctly, the second bounce should be synced with the race start and you should get around +300 burnout wheelie. However, the game often doesn't count the wheelie and you won't get any nitrous. If that happens, restart (if it's IL).

There's also advanced wheelie for 6 gears Corolla, but I haven't tried it yet. It can supposedly reach burnout bonuses of over 600.

Wheelies are quite random and they are also affected by your FPS (the lower the better), grip and gearing. It also takes a bit of practice, so don't give up if you can't get it after just a while.