You can edit your email notification settings in the app settings (View) - 'Email Management' section - 'Email Notification Settings'.
The edit notification screen has many options described in other sections.

You can enable notifications for each account using the slider button beside the account name. Tap on the account name to see the account notification settings, such as sound, filters, schedule etc.

For the fastest notifications, turn on 'Use push notifications (quicker)' in the account notification view. Then select the 'Quick Notifications' option below that setting.

You can choose the 'Notification swipe actions' which will show buttons when you receive a notification when not using AltaMail.

The App badge will display the number of unread emails in the selected email view. By default this is Favorites, but you can choose another panel such as 'Locations'.
To reset the badge counter each time you exit AltaMail, turn on "Clear on exit". This will then show only the new email count since you last viewed the email folder.
If you ever see that the app badge is not correct, make sure you have selected the right email panel view with the badge setting.

You can choose what content you want to display inside notifications. Tap on "Notification Content" to choose which email fields you want to display. eg. Subject, from, from address and email message.