The "Golden cup" icon represents your character's experience; his age, time spent on adventures and so on. Experience is required to progress through missions, unlock new hunt types, and eventually move to another year - the next chapter in character's story.
The "Red fist" represents your character's rating. It is the sum of your equipped outfits, weapons (including all their upgrades) and your available dogs. Spin historical crates or buy new weapons (and upgrade them) to further increase your character's total Rating.