How to measure wave direction in 2D?

If the colours indicate different displacements from the equilibrium position then the locus of all points which are in phase (same colour) is called a wavefront.
The direction of motion of a wave is at right angles to the wavefront.

You should ascribe numerical values to each color (or, better, use the actual numerical values from the software used to build the image) and then perform "numerical differentiation" (Google it) to calculate the gradient of the function - there are several methods, and it is not easy to get decent accuracy. For what it's worth, to get smoother results, you may consider your numbers as a real function of a complex variable (x+iy) and calculate the gradient as a complex numerical derivative of the real function with respect to the complex variable averaged over 4 directions (along the abscissa, along the ordinate, and along the two diagonals).

Generally, it's impossible to define a wave direction in a specific point, as there may be two (or more) superposing waves passing through any point, each with its own direction.

Moreover, it may be impossible to tell how the wave will move, knowing only the displacements at a specific time (though that may depend on what is the wave equation) That is because in most systems, the wave equation is of the second order in time, which means that to solve it you need not only the initail displacements, but also the initial time derivative of the dsplacements.

Assuming that you know the time evolution of the wave, you may be able to identify the set of points at which the displacement has a specific value and analyze how these sets change. But as I said before, it's impossible to calculate from a still image of the wave.