How do I forward my old Outlook.com emails to a different email or to a file at least so that I can access the information in the future? (My Outlook account is being closed by my employer.)
Use a desktop email client such as Thunderbird/Outlook. The client will fetch your emails for offline use and then you will be able to save your data from the program.
An alternative for this is to have a Google mail account which will allow you to import your old emails directly into your new Gmail account's inbox.