How do I use URL aliases in menus?
Like Berdir mentioned in the comments and Greg mentioned in his answer, the following methods of adding path aliases should Just Work™: 


Adding it on the node submission form during node creation/editing
Using Pathauto to automatically create aliases for nodes.
Adding aliases manually by going to Configuration &rarr; Search and metadata &rarr; URL aliases


That it's not doing that is weird. Try clearing your cache by going to Configuration &rarr; Development &rarr; Performance. This should rebuild Drupal's menus and get the path aliases to show correctly.