Using the REST API is probably a good venture when your instance wants to communicate with other servers and systems.
Very important things are to be taken into consideration
1) you have tried to use the API explorer on your civi instance on WP to fetch and display contacts in JSON, from which a request URL will be provide for performing the same operation through REST
2) The site key is replaced in the REST URL with the site key in the civicrm.settings.php probably in ".../wp-content/plugins/civicrm"
3) The security verification process is validated. This is done by ensuring that the current user who sends the request has an api_key to authenticate the request. you could verify this by searching the contact on your civi instance, getting the id, and then querrying you database with " select api_key from civicrm_contact where id='user_id;'
If the api_key is not there update the record with " update civicrm_contact set api_key='chosen_key' where id='user_id "
with all of this you should specify how you want the server to return the data probably in JSON ( by adding argument in URL &json=1 )
Finally you should have a URL that looks like this
The returned data should be useful you can now manipulate and display on the other server page.
You can find more at CiviCRM REST API.