Easy method for the power-up achievements is to just stop at a power-up pick-up location and keep pressing A. It will launch the power-up and pick up a new one right away. Doesn't work for all power-ups, as you need to be ahead of an opponent for some. But I had a hard time getting 3 horns in a race and doing it this way was really easy.
"Life is complicated. I killed people, sold people, smuggled people. Perhaps here, things will be different" - Niko Bellic