Settings > Notifications decides what reaches your bell and your phone.
POP-UPS: Always on, Follow my status (quiet while you're Off the road) or Mute all.
ONE BY ONE: each alert has its own switch (bell and phone together), grouped as Travel (Flight alerts, Weather alerts), Messages (Crew chat, When someone tags you, Direct messages, Group chats) and People (Birthdays: the day before someone on your tour has one, never your own; Friend requests; Trivia challenges).
EMAIL: alerts never come by email unless you turn on Email me what I missed: one email in the morning (about 8 AM Central) listing the alerts you didn't see in the last day, of the kinds you have on. Nothing missed, no email. Turn it off on the same page. A few account emails always come regardless: being added to a tour, someone asking for access to your tour, and support access to your account.
ONE CHAT ONLY: to quiet a single tour's crew chat, use Mute chat inside that chat. Tags still reach you there.