Can I move existing tags, which are currently in a global tag group, to new the new tag groups I've set up?
There is no way to do this natively in Craft through the control panel.

You could either write quick plugin that programmatically moved the existing tags to new tag groups or write some custom SQL that did it directly against the database.

For the latter, it should be as simple as updating the groupId column of the craft_tags table with the new groupId for the tags you want to move.