Every respective service has their own Documentation on how to set up a Transactional Mail into your WordPress. It’s hard to fully describe how to configure these settings and we strongly recommend to read their Documentation and also have a good understanding of how your Hosting Provider works.
There are many Transactional Mail Services available. Some of them will work through the SMTP settings, while other also has the option to work through its API.
Remember that some Transactional Mail Services also highlights the importance to insert some settings into your Domain Records.
See some of the lists to work with Transactional Mail
If you want also to customize your email, there are some services that allow you to customize your email@example.com
Some Services may offer both options, while others only work as Transactions per Server Domain without Mailbox.
You may also find some of them as a WordPress Plugin. Doing a simple search you’ll find some of them available.
REMEMBER to include your customized email into Theme Options > Email Management
and also on your WordPress Settings. Go to Settings > General