Off-screen the Casters are busy developing Chaldea and scrying for the next Singularity or micro-Singularities, the various Servants are raiding those micro-Singularities for resources or preparing for war, and those with worthwhile skills strategise and make plans for the various Servant rosters. Archer also cooks and runs a bar, While Marie helps run errands. Waver is basically on the command staff (a special consultant like Leo) and teaches magecraft to the twins on the side - which is actually the topic of Chapter 14, because with Case Files out on anime I have a better idea of how the curriculum works.
The only people who don't do anything are Gilgamesh, who considers his presence a well-deserved holiday, and Stheno, who is a Goddess.
Meanwhile Ritsuka manages administrative affairs with Roman when he isn't planning for the next Singularity, and Kana handles the Servants and deals with their drama when she isn't planning for the next Singularity.