Teach first asp.net 4.6 (c mvc 5 and webapi), and then asp.net core. For while the bulk of the projects on ASP.Net 4.6, asp.net core until the technology is young, and in production very little. At some points the old and the new ASP.Net very different(for example, the initial configuration of the application, middleware, DI, EF Core), but knowing the old, not difficult to move to a new.
well, main website https://docs.microsoft.com/en-us/aspnet/core/
the most fresh and relevant documentation