The best way to move a contact to a different group is to Search for them.
Once you have found the relevant contact open up their contact record. Here you will be able to see the further options available to you and you will need to select "Groups".
So now you will be able to see the groups that contact is in and it will enable you to amend this really quickly and easily.
All you will need to do is click "Remove from Group" for the one you wish no longer wish the contact to be in and then select the relevant group from the drop down menu and click "Add to Group" for the one you want to add them to.