It is generally better to use "Open Connection" than "Quick Connect" because there are fields to fill out for protocol, username, and password.  If a user uses "Quick Connect", they tend to not specify the protocol or connect as the wrong user.  "Open Connection" also allows you to choose an SSH private key which allows you to connect without typing in your password every time, or to connect to an ssh server that does not allow password authentication.

Once you are connected to a file server, right click on files and select download or upload.  Right clicking on a file within a folder and selecting upload will upload to that folder.  Right clicking on the folder itself will upload to the above directory.  I recommend right clicking to upload and download because dragging files in between Finder and Cyberduck makes it easy to slip and transfer files to the wrong place, or start too many transfers if your first attempt is unresponsive and you try again.

You can leave your home directory by clicking the drop down menu at the top middle which says "/home/uniqname".  You are able to read and write to files your user normally has access to, so you can access /tmp, /scratch, and /nfs this way.