C# is the nicer language (although AS3 and C# are similar) and has a wider range of uses than Flash but since you mentioned games only I would tend to say start with ActionScript 3.0 then move to C#.
Firstly we can ignore the fact that C# is useful for writing business apps since that is not relevant to this question.
C# offers game programming via XNA, SilverLight, and Unity3D compared to ActionScript 3.0 with Flash.
So why do I recommend AS3 to start with? Firstly 2D games are where you should start off and I think this is where Flash excels. The development lifecycle of 2D games is much shorter and tend to produce better gameplay. A lot of indie 3D games suck since they fall short of realising what they started out developing. It is relativly simple to get a 2D game up and running and there is a lot of game engines, physics engines and tutorials available. It is also the platform of choice for artists (that can make all the lovely looking levels, characters etc). Second, the flash platform is the most accessible to your audience. No point making a great game that people never play. Third, AS3 is faily simple to come to terms with.
From my experience the web games that I come across that I think are noteworthy tend to be made in Flash. Checkout Scary Girl and Machinarium.
For 3D or anything that you need the power of the GPU then C# with XNA or Unity3D is what you should be looking at.
I will add that if you are interested in very fast action style 2D games then avoid Flash.