Volcano Manor combines a faction-style questline with Mt. Gelmir exploration and the route toward Rykard. Treat the contracts and the legacy-dungeon progression as related but separate threads.
Joining the Manor
Once you reach Volcano Manor, the key early decision is whether to accept the invitation and begin the contract route. Speak with the Manor residents before rushing deeper into the area.
Contract progression
The assassination requests form their own sequence. Finish the contract you are currently tracking before assuming the next one should already be available.
Talk to the residents
NPC dialogue inside the Manor changes as contracts progress. Revisit relevant characters between steps rather than treating the quest as a pure map-marker checklist.
Rykard timing
The Rykard encounter is tied to the Manor's broader progression. If you care about finishing Manor-related NPC content, check your outstanding conversations and contracts before closing the route.
Use a separate quest note
Record the last completed contract and any NPC dialogue you still need to revisit. This is easier than remembering the whole Mt. Gelmir sequence at once.
Reference: Elden Ring Wiki — Volcano Manor.