The object models are relatively similar. If you understand how inheritance and polymorphism work in Java, things aren't too different in C#. Most of the features that make C# different (structs, operator overloading and anonymous functions, to name a few) are not essential features, so you can write C# code that looks a lot like Java code.
On the other hand, the .NET Framework doesn't have much in common with JRE libraries. So if you learn how to make GUIs, draw graphics, deal with dates, etc. in Java, that knowledge won't transfer. I wouldn't worry about it, though -- being comfortable with the language itself is more important.
I wouldn't pay to learn C#. There are decent tutorials on MSDN, and if you don't like those, there are plenty of free resources on other websites.