All comes down to what your ultimate goal is with programming.
If you look on a job-board like Dice, there's TONS more work available in C#/.NET than regular C++, so if you're looking to make a living developing software that's something to consider.
If you want to focus on 'Desktop Development' learning C# will make things a lot easier. Or you could learn C++ with MFC (Microsoft Foundation Classes) which takes a lot of the headaches out of the way. C# beats C++ hands down for quickness of being able to develop windows apps, however you need the .NET Framework installed to run your apps and comparing to C++ there is a speed-difference (but not an 'OMG THIS PROGRAM IS CRAWLING!' difference

)
If you want Develop Web-Based Applications there's naturally Java, but things like ASP.NET (which uses either VB.NET or C# in the back end) are VERY popular right now together with things like SilverLight & Ajax.
If you're more into the 'nuts & bolts' of how things work, the 'insides' of a program, you want to optimize algorithms etc etc, and don't really care about writing full windows applications but more stuff that's running in the background C++ gives you easier access to everything that's going on there. This can also still be done in C#, however as my post about string conversion in the 'encrypt.exe' thread shows sometimes you need to do a bunch of extra work to do things that might be simpler in C++.
(while the reverse also goes for UI things.. C# is simpler than C++ unless you use MFC etc)
When I started C# didn't even exist, so I had started with Basic on a Commodore 64 (yeah.... that's right lol one of those 'antique' brown boxes)
did Basic on the Amiga, learned some C on the amiga switched to PC, learned some pascal, switched to C++ (
Borland 3.1 in MS-DOS FTW!) since that just felt a lot better to me than pascal and been working with it ever since. (and some TASM too..) Learning C# after already knowing C++ is definitely an easy transition.
That said though if your goal IS to pursue software development as a career then ANY Moden Object-Oriented language that 'clicks' with you mentally and keeps your interest is a PLUS. Just see where it takes you and figure out what feels right.. heck.. some people are making hacks in Pascal/Delphi!
Just figure out what your goal is, and maybe try a tutorial on both and see what feels better to you.