Pieces and monuments of different classes are the main materials required to perform Ascension on your servants which is the process of raising their level cap and gaining new skills.
Farming Pieces & Monuments
In the NA version right now, it is quite difficult to obtain the particular pieces or monuments that you need for your servants as the daily quests are a mix of servants of various classes, unlike the JP version which only has one class per day.
The most effective way of farming is by doing the 40AP quest as you would get at least one chess piece per round, but not guaranteed for monuments. The schedule for the daily quests is listed as below:
Besides doing the daily quests, you also have a slight chance of obtaining the pieces by doing the free quests that have servants as the enemies. Here is the list of it: