First, there is no such thing as MVC 6. It's ASP.NET Core. Second, the decision of whether to use .NET Core or .NET Framework comes down to what you need to do. .NET Core is cross-platform (Windows, Mac and Linux), whereas .NET Framework is Windows-only. .NET Core is also lighter weight, faster, and has a number of new CLR features, compared to .NET Framework. Virtually the only reason to use .NET Framework is if you have a dependency on something that utilizes Windows-specific APIs, meaning it can't be run cross-platform. Otherwise, you should always use .NET Core, if you can get away with it.
ASP.NET Core is really just a collection of NuGet packages, and like other NuGet packages, they can be updated easily. However, the difference is that these NuGet packages are generally tied to a particular version of .NET Core, and thus, the appropriate version of the packages are installed based on which version of .NET Core you're targeting. For example, if you're targeting netcoreapp2.2 then you'll see that that packages like Microsoft.AspNetCore.App will be brought in with versions like 2.2.x.
So, to answer your question about staying up to date, it essentially entails installing the latest version of the .NET Core SDK, and then changing your project to target that new version. Then, all the ASP.NET Core NuGet packages will naturally update accordingly.