Red is a primary color and you cannot create red by mixing paints together. You can, however, change the color of any red paint and you can make a red paint look brighter by pairing it with particular colors.
As much as you may wish to, you cannot make a red paint brighter or more saturated than it is straight out of the tube. Instead, you need to choose a red paint based on the desired results. Since it is a primary color, there are a good number of red shades available in almost any paint. Among the most popular are the cadmium red and vermillion. You'll also find earthy reds such as the popular burnt sienna.
If you mix red paint with other pigments, you will begin to get different hues. Mix a yellow into it and you'll create an orange-red. Mix it with titanium white and it will begin to turn pink, but mixing red with zinc white will reduce the saturation. If you mix a red with blue, you're heading towards purple.
Red is a very useful paint in your toolkit and the color possibilities when mixing with it are endless. Yet, you must always remember that you cannot make a red paint "redder" than it is already.
There is a little trick that you can use to create the illusion of your red being brighter. It all depends on the colors and tones that you paint next to it.
The complementary color of red is green and this is the perfect place to begin. Complementary colors naturally make each other appear to be brighter than they are in reality.
To see how your red appears next to other colors, take a few minutes and paint up a color chart with blocks of red surrounded by various colors. When you're done, examine it to compare the results. You should notice a significant difference in how the red pops out from the various tones. This can guide you in how to apply the reds in your painting for the desired results.