A "bright color" can be two different things. One is a color that is brilliant and arresting, like magenta or citron yellow. These two colors are bright because they are fully saturated; that is, they are pure colors with no mixtures of a neutral or gray color to tone them down. This kind of bright relates to a component of color called chroma. The more chroma, the brighter and more intense the color is. The other brightness meaning relates to value: how light or dark the color is. Colors higher in value are closer to white. To get the brightest colors for your painting projects, learn about two different brightness strategies and choosing according to your intent.
Get the very brightest colors, in terms of chroma, by using the colors right out of the tube and not mixing them at all. The principle of paint is that every time you mix a color in, you lose intensity, because the color you add will absorb light that the other color would reflect, producing a muddier tone. So keep the colors at their brightest by selecting brilliant colors and using them directly on your surfaces. If you need to mix, choose colors that are next to each other on the color wheel for best results, like green and yellow, or blue and purple.
Mix bright value colors by starting with white on your palette or in your mixing bowl and adding another color to that a little at a time. Other colors will be stronger in pigmentation than white, so you may not need very much to get a bright pastel that looks colorful. Stick with just one color for best results. With enough white you can make even a color that is naturally dark, like blue, appear bright. As before, if you need to add a second color, add an adjacent color from the color wheel.
Avoid complementary colors and black when mixing bright paint colors. Complementary pairs, like red and green, will cancel each other out and turn your paint brown or gray when mixed. Black, aside from being the darkest paint color, often has a blue or brown cast that will create off colors when mixed.