To rename an existing page, you require 'Edit Pages' permission which is assigned by a space administrator from the Space Administration screens. See Space permissions or contact a space administrator for more information.
Confluence will automatically update all internal links to point to your new page title. This only works for relative links that use the relative link syntax, for example [Renaming a Page].
In macros, page links will not be updated. You need to update these manually.
Links from external sites will be broken, unless they use the permanent URL - see Working with Links.