Using Profiles to allow member to view/edit own details only
Maybe this is what you're asking for:

Create a new profile. Call it something friendly for the user like "My Contact Details". Tick it as a "Standalone Form or Directory"
Fill it with the fields you want the user to edit.

Pick the page you want them to be accessible and edit it. Use the CiviCRM button to pick the profile or enter in the shortcode yourself.

It should similar to,

[civicrm component="profile" gid="123" mode="edit" hijack="0"]


When a user is logged in, this will contain the fields you selected to be edited by the user.

Edit: while the anonymous visitor will not see the profile fields until he/she is logged in,  I guess it'd be good practice to make the wp page private to avoid any slips and keep the design fluid  - or use a plugin to limit the page to logged in users.