1- Contrasting colors
Remember the color wheel from high school art class? Colors that sit across from each other on the wheel such as blue and orange, yellow and purple, and green and red are called contrasting colors. When mixed together, contrasting colors become more grey-like and dull, but when placed next to each other, they become more vibrant. In photography, look for subject matter that naturally has has high color contrast, or set up a scene to add more contrast.
2- Bold colors against black or white.
Using subjects containing bright, bold colors against a more muted (mostly black or white) background is an easy way to make your subject pop out from the photograph. The eye will automatically be drawn toward the bright color making the subject immediately noticeable.
3- Bold colors against a uniform background of another color.
Similar to the second one, this technique involves photographing bright or bold colors against a uniform background of a different color.
In order to make the colors in photos contrast even more, you can use a photo-editing program to play around with the shadow and highlight levels. Adding more shadow makes the dark part of a photo even darker, resulting in the contrast between the shadow and bright color becoming more noticeable. Brightening the highlights in a photo will have the same effect. Some programs will allow you to saturate the color in the photos, making less intense colors more bold.