Where can I see the emails sent to and received from my customer?

You can see the emails sent to and received from each customer using the Touchpoints feature on a contact record.

Clicking the Touchpoints button will display the Summary screen where you can see metrics for different touchpoints.
The Email section displays the number of emails sent and received for a contact. Clicking on any of these numbers will display the details in a table format 

To know more about Inbox, click here.