We recently moved our on-premise TFS server to a new server and we are able to access our projects on the new server.
However, I am unable to see the option to add a new project, only to browse to existing projects:
I have checked my permissions and my permissions include:
This should give me enough rights to add a new project. I'm sure I am missing something, but I cannot see what it is.
In Visual Studio Team Services (formerly Visual Studio Online) you can create your projects from the web interface.
But in TFS you need to fire up a matching version of Visual Studio (in your case 2013), and create new projects from the Team Explorer window.
From Team Foundation Server 2015 update 2 onwards you'll be able to create team projects using the web interface. With all older versions of TFS you need to follow these steps.