User Profile Self Update
You have two options for this.

One is to use CiviCRM profiles.  There's an instructional video and a section in the profiles documentation on using profiles as a form.

You can also use Webform-CiviCRM integration (if on Drupal) or Caldera Forms with CiviCRM integration (if on WordPress).  These will give you more flexibility on the look and feel, and are generally more powerful - but from what you describe, you can use either.