The first step of this puzzle is taking the cue from the priest that the exact colors are relevant. Loading the stained glass window into a graphics editor (such as GIMP, available on Athena), it's possible to use a tool to get the exact hex color of each pane. Each pane is pure red, green, or blue, so that only two digits of each are non-zero. The colors of the panes, and their sums, are as follows.
The discussion of "changes," the fact that these are all strings of six digits which are all 6, 7, 8, or 9, and the eight by eight table headed by a father, mother, and six children all suggest the I Ching, whose trigrams are associated with the eight members of a family. 7 stands for masculine, 8 for feminine; 6, feminine changing to masculine, and 9, masculine changing to feminine.
Each string of six digits can be taken to be a pair of hexagrams. For instance, the hexagram in the first row, 778676, can be taken as a move from 778777 to 778878. The pairs (not in order) are therefore:
Taking the floor tiles as a grid, then, and labelling each row and column with the trigrams, one can trace lines (just as the boss paced them out on the floor) from the first hexagram in each pair to the second. These give (more or less, depending on how square your grid is):
and the answer is MOOD.
The training is M TO F CHANGE, and this gives the new answer, FOOD.