Right now I am currently working with Outlook 2013 that is installed on my workstation. I automatically get signed in. I do not need to supply my user name and password.

Right now I am new to SharePoint 2013 and I need to be able to see email messages that are sent to 3 different test email accounts when I am testing the SharePoint changes.

Thus can you tell me what I can do so that I can access the 4 different email accounts (my account and 3 test email accounts) when I am testing the SharePoint changes. I definitely would like to know to know how to access these different email accounts.

You can either add those accounts to your current Outlook profile, or create one or more new profiles for those accounts.  See:

Add or remove an email account
https://support.office.com/en-ca/article/Add-or-remove-an-email-account-5a74e3be-7e11-4d9b-99f7-08ec25e9d072?ui=en-US&rs=en-CA&ad=CA

Overview of Outlook e-mail profiles
https://support.office.com/en-ca/article/Overview-of-Outlook-e-mail-profiles-9073a8ac-c3d6-421d-b5b9-fcedff7642fc?ui=en-US&rs=en-CA&ad=CA

I followed your link for the add or remove email addresses and tried to add the test accounts. The connections did not happen. I then looked at the directions for the advanced options and did not understand what some of it meant.

I know the email accounts exist since I can send them an email message and I did not get the message saying the email account is invalid.

Do I need to setup profiles first? then add accounts

What do you mean by "the connections did not happen"?  Do you see those accounts listed on the Accounts Settings page?

You can add an account to an existing profile or create a new profile to add the account to - it's really up to you.  You have to choose which profile you want to use when you start Outlook if you want to access a profile that has a specific set of accounts.  You can control whether to use a default profile or to be asked to just a profile by configuring this option via the Mail applet in Control Panel.

Let me know if you are still having problems.