Learn how the color wheel works. The color wheel is based on the three primary colors, red, blue and yellow. They form a Y shape and the spaces in between them represent the colors you get by mixing them together. For example, mixing red and yellow gives orange. These colors are known as secondary colors. Mixing adjacent primary and secondary colors gives tertiary colors.
Know your complementary colors. Complementary colors are the colors directly opposite to each other on the color wheel. These colors bring out the best attributes of each other and always match. In the case of red, its complementary color is green.
Try analagous colors. Colors located in one "pie slice" of the wheel are called analagous colors and work well together. For red, this could be several shades of red, or simply red, orange and yellow.
Use a triad relationship. This means three colors that are evenly spaced from each other on the wheel. In a simple color wheel example, that would be red, yellow and blue.
Combine split complementary colors. This means matching one color with the colors on each side of its complementary color. For example, red could thus be matched with light green and aqua blue. This kind of combination is softer than using complementary colors only and provides harmony.
Use neutrals. Neutral colors like black, white and grey provide contrast but allow the red to be the focus. Neutrals will work with any shade of red.