I am trying to create a simple script that displays who a user follows and if that person follows them back or not. Is there a way to fetch this information with Instagram API?
These are the relevant Instagram API docs you should refer to for specifics.
As a rough overview: you will need to register a developer account, create an app, and authorize that app to receive an access token. Once you have that, something like this will do what you want:
$userId = "self";
$url = "https://api.instagram.com/v1/users/$user/followed-by?access_token=$accessToken";
$response = json_decode(file_get_contents($url));
$followers = $response->data;
Of course you can also use cURL or Guzzle instead.
Note: to get a user's follower list that user must authorize your application with scope=follower_list to be granted this permission.