Every online booking sends you one email, with a link to the diary and a Message button when messages are on. It goes to the address in New booking notifications go to, under Shop settings, Keeping customers, Notifications; leave it blank and it goes to your business email. Replying to it writes to the customer. Each member of staff can also get their own copy of the bookings made with them, if that switch is on their card under People and places; it goes to the address they log in with.
The FullDiary app adds a notification on your phone the moment someone books online. It also tells you about an online cancellation, a message from a customer and a card declined at the desk. Staff get the ones their login can see. In the app, open More, then Account & plan (Your name and notifications, if you do not see that), then Notifications, to switch any of them off on that phone. Your phone's own settings need to allow notifications from FullDiary too.
A booking you add yourself in the diary sends no alert, because you typed it.