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Should I choose to learn Java or .NET?
Is there an advantage to learning Java or .NET first?
(ie. would the transition from J2EE to .NET be significantly easier than the reverse?)
Do you think that one platform has overtaken the other in terms of industry use?
(feel free to be as biased or as objective as you like)
see also: Java or .NET?
What kind of jobs do you prefer?
.NET is (mostly) windows-only, and Java is on all platforms. There are quite a few companies which use Linux as server, for which .NET would be out of the question.
On the other hand, I have the feeling that C# (.NET is a platform, C# is a Java-like language) is a bit further in convenience for developers. The stack is a bit more integrated (with other Microsoft solutions at least).
I'd say neither. Java is not better than .NET, and .NET is not better than Java. Of course, we could discuss usage, strengths and weaknesses, technical maturity etc., but that would have no value to your question.
Learn the one you find yourself most comfortable with, and/or need or want to to know where you work/study. What's important is knowing the way of thinking when programming, not the languages themselves. They can often be taught nearly on a per project-basis. If you're new to programming, you won't notice any huge differences between the two for a while anyways.
Amount of jobs: A lot within both. Last time i checked where I live (Norway) it was roughly 60% .NET and 40% Java.
Having done both, I think it's easier to move from Java to .NET and likewise, .NET is easier to learn than Java. Both are great languages. However, I think the .NET stack is easier to learn because there are fewer choices you have to make since Microsoft provides the entire stack, and because the .NET API is a bit more consistent than Java's.
I don't really see an advantage as far as which to learn first. I think go where your interest lies.