Redshock's got some fair points but I think they could be framed a little better...
I would suggest C++. If you're wanting a job later
... in the AAA games industry.
Yes that's true, the games industry is fueled on C++ by and large.
Having said that, the OP did highlight that he already is employed in QA and is looking to do this as a hobby not as a job. So these are maybe not things the OP need worry about right now.
dropping into unmanaged (native) code for writing performance enhancers where it'd be slow if written in Java/C#, and just being better aware of performance issues (C# and Java's runtimes were built using C++).
Absolutely, there's no question that knowing C++ is beneficial to a developer for these (and other!) reasons.
At the same time to be a well-rounded and knowledgeable developer it is of the utmost importance to know other languages too such as Java/C#/Python/Scala/F#/javascript/etc. And it doesn't really matter which order these are tackled in.
Those who specialize in C#/Java tend to be limited as sandbox users, though that's fine depending on the project.
That is somewhat true depending on how you define 'sandbox' but it's not very limiting in practice unless you're writing something low-level like a driver or operating system.
Even C++ has a logical VM 'sandbox', defined by the standard, and this does not map directly to the underlying physical machine architecture either.
If you're just doing this for a hobby and don't plan to do game development or software development, like many here enjoy indie/hobby work, Java is a good choice, and C# is closely related and better coupled with the Windows operating system
I think this is the real take-away point that is on-topic for this thread.
I would just temper it by pointing out that there are many software development opportunities out there for languages other than C++. I myself currently work as a back-end Java developer by day.
C# [snip] has a better JIT compiler for performance.
Actually Java's Hotspot JIT uses a lot of advanced techniques to decide what/when/how to JIT. The JIT compiler in .Net is actually comparatively basic. This is an area where Sun/Oracle have invested a lot more time than Microsoft. Microsoft have spent that time building the language itself and the tooling around it. So it's horses for courses really.
While C#/Java hobbyists might not like the quote, it's not meant to offend, but it's what at least one recruiter looks for.
Of course! No offence taken. It's just a viewpoint. It's one I happen to disagree with but no reason to take offence all the same.
What I see is actually a very good demonstration that recruiters don't know what they're talking about when it comes to programming. I would not what that guy doing my company's recruitment.