Choosing a version of .NET (Framework or Core)?

I've been learning ASP.NET MVC and would love some input on different versions. My place of work utilizes Framework 4.5.1, but it seems that is pretty outdated by this point. However I'm a bit confused by considering a somewhat outdated Framework version, the newest Framework version (4.7.1), and the newest Core version (2.1). Obviously it's important to be familiar with my employer's existing systems (and therefore Framework 4.5.1) but for future systems, is there a particular 'right' or 'wrong' choice for versions to use/learn? Are the differences between each of these huge, or is it actually not that big of a deal?

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This and any other info on .Net Framework and .Net Core versions is super helpful. Thanks!