How to convert a dot product of two vectors to the angle between the vectors.

The dot product of two normalized vectors is equal to the cosine of the angle between them. In general $$\cos \phi = \frac{a\cdot b}{|a||b|},$$ since your vectors are normalized, $|a|=|b|=1$ and $\phi = \arccos(a\cdot b)$

Because you want an answer in the range [0,2pi) and not [0,pi], I think that the question's title is misleading. I think that you are asking, "How can I calculate the angle that will change the direction from vector a to vector b?"

I would not use the arccos formula for dot products, but instead use the arctan2 function for both vectors and subtract the angles. The arctan2 function is given both x and y of the vector so that it can give an angle in the full range [0,2pi) and not just [-pi,pi] which is typical for arctan.

The angle you are looing for would be given by:

arctan2(b_y, b_x) - arctan2(a_y, a_x)

The result may be a negative angle, but at least it will go from vector a to vector b. If you want only positive angles, then add 2pi when the angle is negative.

Beware that arctan2 takes y then x. This is to be similar to arctan which takes y/x as input.