I was wondering in a situation where the user is manually approved (no CBSubs installed) that the user can be automatically moved to a different user group using Auto Actions?

You'll probably want to use the after approval trigger. Is there a reason you need to assign a new usergroup on approval though? They can't login until approved so usergroup has no meaning to them yet and it's probably better to just give them the usergroups you want during registered (within User > Manage > Parameters you can set the registration usergroup).

I am unsure which TYPE I should be using?

The Usergroup type.