Which is the quickest way to change the font color in multiple, randomly located cells in a worksheet?

You could use the Format Painter, double clicking on it before it starts while selecting a cell that has the colour you want. You could also select the cells first, by clicking on them while holding the Ctrl key and then choose the colour you want from the font settings.

First select or highlight the rows or columns that you want to change the font color. You can hold the ctrl key while select multiple cells. After, selecting you can now change the font color of everything you selected.