Is it possible to automatically email individuals once I manually add them to a group?
You are able to use Scheduled Reminders (Administer > Communication > Scheduled Reminders) to automate messages and set when they go out. For your goal, you could limit the messages to go out only to a specified group/mailing list and you can based the automation on something such as when the contact record was initially created. 

Click "Add Reminder", give it a title, choose entity type "contact" and "created date". Under "limit or add recipients" choose your mailing list. Then set up various messages and times based on creation date.