When I started writing this question, my problem was that after a successful VSTS Build, I wasn't able to see the files relating to my web application project for release. Only the files from certain other projects in the solution were present. However, I just came across this question, which has helped.
I can now see the compiled .dll files for my web application project, after altering the configuration of the Content setting in the Build - that is, the contents of the Bin folder under that project. But I can't see anywhere the other files I need to copy the built web application to my server - the views, the scripts, the css, etc.
I'm finding the power and flexibility of VSTS's Build and Release functionality very confusing as it's complete overkill for our requirements. Up until now, I've just right-clicked on the web app project in Visual Studio selected Publish and used the File System publish method. Easy. Now that I want to automate the building and deploying of the application, it's many times more complicated!
So, can anybody tell me how I can get the solution to build in VSTS in such a way that I can then use a Copy Files task in the Release Definition to copy the files to our web server (the server isn't visible to the Internet so I'm using a locally-hosted Agent)?