You can easily do this in quick play, although there are plenty of chances during the story playthrough.
This was a bit confusing for me at first, as I thought my needle was green right from the start. The needle displaying your speed/rpm can be yellow, green or red. Yellow means to early to shift, green is the right time, and red is too late. There is a shift button on the top right. More or less once the speed/rpm on the bottom left of your screen is in between the white and the red part, it will turn green for a short while. Tap the shift icon once the needle turns green. Do so for a few more times until the gear is in the 6 (bottom right), which is the maximum. Then press the nitro and finis first. you don't have to worry about steering.