To advance a character's route, they need to be available/free for the day, and then you can select them in the menu/list with the 'plan for the day'. (If done right, the plan for the day should display your choice.)
This means no assigned jobs for them that day. If they are assigned to research or the valley, take them out to make them available.
Some additional notes:
- Injured staff are unavailable
- Some staff can only become available after certain time-gated events (e.g. Asterion can only be available after some event in later days)
- Some staff don't have routes yet
- It's okay to have nothing to do that day if you really want to devote your staff in doing research and the valley. I don't recommend this though.