Sections of Soul Trap unlock after completing each of the first 6 towers. For those familiar with the old system, you'll be happy to see all of the stats in Soul Trap are available as you complete the towers. You do not need additional materials to unlock each individual stat in each section here. Also, re-rolling stats takes Stones of Destiny instead of Diamonds as it once did. So we can actually make practical use of this without spending. Stats here work as everything else in the game does. White/Blue/Purple/Pink/Gold is the order of stat boosts. White stats being the smallest boosts you can get, Gold being some of the highest stats you can get for that section.
Note: I'm actually not positive if there are White stats, but you get the point.
TIP: I would recommend focusing on upgrading the Zodiac first since you know exactly what stats you'll be getting. The stat boosts here are randomly generated so you could waste a ton of Stones of Destiny and not actually see any improvement. Just stick with whatever you initially unlock and get to this much later.