Just to clarify, you need to upgrade all the armour pieces of the same set, not just one of each type.
The easiest set for me to do was Surtr's set.
By doing the Trials of Muspelheim (needed for another trophy anyway) you unlock both the armour pieces, and the materials needed for the upgrades. You get the armour pieces at level 7, so you only need 2 upgrades per piece in order to reach the maximum of level 9. Plus you get a toooon of Hacksilver so no need to worry about those costs either.