Once you own a color card, tap on a ride, restaurant, shop or coaster to open its Information panel, then tap on the Color Icon (the color wheel) to enter the Color menu.
You can chose which part of the ride you want to re-color by tapping on the categories on the left side. For rides, restaurants and shop, the sections are: Main, secondary, and Detail. For coasters they are: Spine, Rail and Support. Note that for now, the train and station will also take the assigned color to the track.
To assign a color to that section, simply drag the color on the park, or tap on it.
You can also go back to the default one if you don't like the new applied color, by choosing the first option in the color list.