How to create groups and assign users to them?
This sounds like a clear use case for the Organic Groups module. Organic Groups is great if you have a need to create a container for content and then associate users with that container.

However, if you don't have a need for any content-oriented aspects of OG (such as access control), then maybe all you need to do is make the associations visible, rather than a part of the site's information architecture. In that case, it sounds like you might want to use the Content Profile so you can leverage CCK Node Reference fields, which can be one:many arrangements, and using the bundled content_permissions module or Field Permissions, restricted to roles with specific permissions for editing the field.