Two options for the past emails:

1) You'll want to store the emails as activities, which you can import from a CSV. I'm sure you can find a Google Apps Script that will pull all a users emails based on whatever criteria you like into Google Sheets and then export that to CSV.
Do you want to connect these activities to other contacts in CiviCRM (the contacts the emails went to)?

2) Depending on the number of accounts you are talking about and how the emails are stored, you could use Email-to-Activity processing with a specified IMAP folder and just go through each email account in turn by changing the settings in the Mail Account.

For new messages, have you read [Email-to-Activity processing][1]?


  [1]: https://docs.civicrm.org/sysadmin/en/latest/setup/civimail/#email-to-activity-processing