There are several add-ons that would accomplish what you're looking to do. Nick already Mentioned Zoo Visitor. Here is a list of other add-ons that will also work well:
Zoo Visitor, Profile:Edit, and Safecracker Registration have the added feature of storing members as Channel entries. So if ...
It is in the Control Panel. If you don't see a design section, it means the user group your member is in doesn't have permission to see the Design section.
Ahh, for some reason I thought you were using EE2. The area of the documentation you mention is out dated. I will provide the correct link to follow to create a copy of the built in templates so ...
Zoo Visitor was highly recommended by other developers on Twitter, and I've been quite pleased with it as well.
It supports member group selection in registration forms (and you can create as many as you'd like). Here's the relevant portion of the docs.