First of all ASP.NET is request/response pipeline. This means that you are given access to the request and response streams as well as some provisions like, session, cache, security, etc.
On top of this there are 3 frameworks in charge for generating HTML. The first and oldest is known as ASP.NET Web Forms. Because it was the only one it is sometimes called ASP.NET but this is not correct in the current state of things. ASP.NET MVC is the second one and there is a third one known as ASP.NET Web Pages. All 3 of these share the same core ASP.NET request/response pipeline and the APIs for Session, Cache... What is different is how they generate HTML.
You can check my answer to this question for more info.
Asp.Net Web Forms and Asp.Net Web Pages
And to answer your concrete question - no Web Forms is not going away. A lot of people use it, MS are releasing new versions.
Web Forms is pretty good for people with desktop background because it uses a control model familiar to desktop devs and also has something that simulates state. It also requires less knowledge of HTML, JS, CSS. ASP.NET MVC is kind of the opposite. It gives you a lot of control but requires a lot of knowledge about the web.
I personally prefer Web Forms to MVC for a variety of reasons that I will not list here but even Web Forms supporters (and especially me) will admit that Web Forms is pretty bad way to learn about the web because it abstracts a lot of things. This gives you productivity, security, etc. but can result in cases of leaky abstraction if you don't know how the underlying framework works and it is pretty easy to skip learning the details because you know stuff just works... until it breaks and then you don't know where to start.
Ultimately the choice is yours but if you start with Web Forms be sure to learn about HTTP verbs, cookies, raw response stream, http headers, html form/submit model inline css vs separate files and javascript out of the context of Web Forms and make sure you know how Web Forms automates these.