In one of these days I would like to start with a new project and MVC seems really interesting, but I was wondering if it is possible to upgrade a MVC 5 project to a MVC 6 project, which will be released later this year?
Or do you have to start all over again since a lot of things have been changed? Or do you recommend me to wait until MVC 6 is released?
It won't be a direct upgrade path, as a lot of breaking changes were made. You can start playing with MVC 6 now though, no need to wait. You can download the Visual Studio 2015 Preview and get started.
Some of your code will move over, but you'd need to go through and verify every bit of it and fix a lot of errors. It's probably better to start from the ground up with MVC 6 (Microsoft has recommended this in some Q&A on videos I've seen).
Edit 11/28/2015
Visual Studio 2015 is officially out now, and ASP.NET 5 and MVC 6 are in RC1 status. You can create a new ASP.NET 5 project with MVC 6, then just copy over all the source files. After that, work though the errors one by one until you've solved them all.
Migrating From ASP.NET MVC 5 to Core MVC
This article shows how to get started migrating an ASP.NET MVC project
to ASP.NET Core MVC. In the process, it highlights many of the things
that have changed from ASP.NET MVC. Migrating from ASP.NET MVC is a
multiple step process and this article covers the initial setup, basic
controllers and views, static content, and client-side dependencies.
Additional articles cover migrating configuration and identity code
found in many ASP.NET MVC projects.
Here is the URL: Migrating From ASP.NET MVC 5 to Core MVC