What is the best way to change the user profile edit page? I need to add a new field alongside the password.
You can add new fields to the users profile page at admin/config/people/accounts/fields
You can also manipulate the form by implementing
hook_form_user_profile_form_alter(&$form, &$form_state, $form_id) and
hook_form_user_register_form_alter(&$form, &$form_state, $form_id)
To add new profile pages you can use the Profile 2 module.
In drupal 7 /admin/config/people/accounts/fields Here you can add and change certain fields for the user profile. You may want to check out the module profile2 as well: https://drupal.org/project/profile2