Global "connection"-like variables in Python

You should create a class (called something like CloudContainer) that includes all of those global variables as members, and rewrite it as (just as a start):

class CloudContainers(object):
    def __init__(self, username, key, originContainerName, targetContainerName):
        self.username = username
        self.key = key     
        self.originContainerName = originContainerName
        self.targetContainerName = targetContainerName

    def cloudConnect(self):
        print "Creating connection"
        self.connection = cloudfiles.get_connection(self.username,self.key,servicenet=True)
        print "-- [DONE]"
        print "Accessing containers"
        self.originContainer = connection.create_container(self.originContainerName)
        self.targetContainer = connection.create_container(self.targetContainerName)
        print "-- [DONE]"
        return

    def uploadImg(self, new_name):
        new_obj = self.targetContainer.create_object(new_name)
        new_obj.content_type = 'image/jpeg'
        new_obj.load_from_filename("up/"+new_name)

    def getImg(name):
        obj = self.originContainer.get_object(name)
        obj.save_to_filename("down/"+name)

Thus, any function that uses these global variables (like getImg and uploadImg above) would be included as a method of the class.

Easier, yes, but it means that it's very hard to tell when and why those variables get changed. I think the best answer is the one you have given in your question - pass them around as an object. E.g:

def cloud_connect(origin_container_name, target_container_name):
    print "Creating connection"
    connection = cloudfiles.get_connection(username, key, servicenet=True)
    print "-- [DONE]"
    print "Accessing containers"
    origin_container = connection.create_container(origin_container_name)
    target_container = connection.create_container(target_container_name)
    print "-- [DONE]"
    return connection, origin_container, target_container

Then simply pass that tuple around.