It sounds like your primary focus is web technologies. In that case, jump straight into .NET Core otherwise you'll be disappointed by the stuff that used to be in .NET Framework that you lose in .NET Core and you'll have to re-invent yourself. (ex. web.config settings, ability to update binaries while web app is running, ability to encrypt secrets in the configuration file, etc.)
ASP.NET is the core technology that provides the ability to process web requests. I think that your question is really about what is the difference between ASP.NET WebForms and ASP.NET MVC. The short answer there is that WebForms was written for Windows developers transitioning into writing web applications -- idea was to hide the stateless nature of HTTP and make it seem stateful like on the desktop, as well as, encapsulate HTML controls and make them act like Windows controls. Unfortunately, the abstraction quickly breaks down and the developer ends up needing to learn about the page lifecycle and how ASP.NET WebForms tries to deliver the stateless abstraction. Additionallly, ASP.NET WebForms tends to work best for the pre-Web 2.0, pre-AJAX style web sites where there's a page reload for every action. ASP.NET MVC doesn't do any of that abstraction -- it exposes the developer straight on to the stateless nature of HTTP and encourages the developer to use the Model-View-Controller design pattern to deal with it. It was introduced at the height of everybody extolling Ruby-on-Rails and its use of the MVC design pattern to make things easier for developers to design and test their code.