I found this helpful article which explains in detail how to assign a newly created use to a specific user group in Joomla. http://www.pages-and-items.com/extensions/frontend-user-access/faqs/assign-users-to-group-from-select-on-registration-form
I haven't tried it yet to see how well it works or how easily it is done.
It looks like it covers off modifying the standard form although a lot of the example code seems to be referencing Community Builder fields.