We want to ensure that we pick at least one ball of each color. In the worst-case scenario, we could pick all the balls of two colors and still not have a ball of the third color. So, let's assume we pick all the green and white balls first, and we are left with the red balls. Total number of green and white balls = $8$ (green) + $7$ (white) = $15$ balls. After picking all $15$ balls, we will be left with only red balls.

Next pick: The very next ball we pick (the $16$th ball) must be red, as only red balls remain. Thus, to be assured of having at least one ball of each color, we must pick $16$ balls in total.

Conclusion: The minimum number of balls that must be picked to ensure at least one ball of each color is $16$.