Is it possible to move some tags from one tag group to another or I need to add them manually and then delete if from exiting group?
Thanks
You could have a look at the exp_tag_entries database table, in particular the tag_group_id column and make manual modifications, it's not recommended without having backups ready and knowing what you're doing.
The best and safest way would indeed be to delete the tag in the Tag CP, then recreate it from an entry form in a Tag custom field assigned to your new tag group. Note: when deleting the tag, this tag won't be associated to any entries anymore.