I am developing a Win8 (WinRT, C#, XAML) client application (CSOM) that needs to download/upload files from/to SharePoint 2013.
How do I do the Download/Upload?
I would suggest reading some Microsoft documentation on what you can do with CSOM. This might be one example of what you are looking for, but there is a huge API documented in msdn.
// Starting with ClientContext, the constructor requires a URL to the
// server running SharePoint.
ClientContext context = new ClientContext("http://SiteUrl");
// Assume that the web has a list named "Announcements".
List announcementsList = context.Web.Lists.GetByTitle("Announcements");
// Assume there is a list item with ID=1.
ListItem listItem = announcementsList.Items.GetById(1);
// Write a new value to the Body field of the Announcement item.
listItem["Body"] = "This is my new value!!";
listItem.Update();
context.ExecuteQuery();
From: http://msdn.microsoft.com/en-us/library/fp179912.aspx