It is possiblet to customize the fields available on the standard Joomla registration form.
Instructional video at: http://www.youtube.com/watch?v=dk_86nHu3AM
While this approach is viable, I believe it introduces some overhead when you're maintaining the site i the future. As a result, you may want to consider using Community Builder or JomSocial extensions to which give you the ability to customize the user experience. Keep in mind; however, that both of these extensions have a tonne of features and they may be way more than you need.