So you have 3 points on the first plane, (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3). They map to 3 other points (s1, t1, u1), (s2, t2, u2) and (s3, t3, u3). You'd like to figure out the transformation to map the rest of the first plane onto the second plane.

The simplest solution is to realize that for any point in the first plane, there must exist a and b such that the point can be written as (x1 + a*(x2-x1) + b*(x3-x1), y1 + a*(y2-y1) + b*(y3-y1), z1 + a*(z2-z1) + b*(z3-z1)). The affine transformation that takes the first plane to the second will map that point to (s1 + a*(s2-s1) + b*(s3-s1), t1 + a*(t2-t1) + b*(t3-t1), u1 + a*(u2-u1) + b*(u3-u1)).

Now given any point in the first plane, all you have to do is solve for a and b, then you've got your answer.

(Please note that the direction that BrowserUk pointed you in usually gives the wrong answers, and can't even be tried if the planes are parallel.)