I think you hit the highlights. Any teacher who attends a one day or longer training gets a user account (publisher level) in the NHEdGIS organization. They can then use these as they take further trainings or to practice on their own. We also have generic accounts that can be used for shorter trainings such as those offered at teacher association meetings.
As far as assisting teachers with setting up their own organizations, we encourage them to take the one day training. If they can't, or if they did at some time in the past and are setting up a new organization at a different school, I volunteer to be on the phone with them when they set it up to help them make decisions. We also strongly encourage them to add either me or another one of our NHEdGIS key people as an administrator in their organization so we can help troubleshoot or regain access to the organization if the people who originally set it up leave the school.
Although this relates more to helping schools set up their own organizations then to the statewide organization, one of the things that is a little strange is that you have to create the first administrator account before you create the organization. (Warn them not to join their personal account or their other organization accounts to the new organization.) Since the standard naming conventions appends the organization short name following an underscore for each user, it makes sense that the first administrator account also follows that pattern, yet it's not until the next step that you find out whether the short name you've selected is available. rlilja figured out the solution to that problem. In a tab of a web browser type: http://shortname.maps.arcgis.com where "short name" is replaced with your desired short name. If this ends up taking you to either the homepage of an organization, or a sign in page, you know the desired short name is not available and you can try again. Once you've found an appropriate and available short name you can then create the administrator account following the same standard pattern as ArcGIS Online will use for the other accounts in the org.
