I used to create all of my table relationships using database diagrams, but after installing VS2013, the feature seems to be missing. Was it removed completely, or can I turn it back on?
Yes they have unfortunately
"In previous versions of Visual Studio, you could use Visual Database Tools to create tables, queries, and views, and to create and maintain database diagrams. In Visual Studio 2013, Visual Database Tools have been deprecated, but SQL Server Database Tools provide replacement features to help you create tables, queries, and views. See SQL Server Data Tools (SSDT).
Neither Visual Studio 2013 nor SQL Server Database Tools provide a replacement for database diagrams. You can use SQL Server Management Studio (SSMS) to work with database diagrams. SSMS is installed with SQL Server 2012 SP1 Express (download center)."