I know about XNA, but while my PC meets the minimum requirements, my friend's PC lacks a sufficient video card so he has issues with it. I was wondering if there were other options outside of XNA for someone learning programming in C#.
I hope I worded that properly.
If you're still "learning programming in C#", I would start there instead of starting with "game programming."
Game programming with XNA requires a decent framework of standard C# programming skills. Once you have the basics down, then you can tackle using something like XNA.
That being said, if you don't meet the minimum requirements for XNA, and you still want to do Game programming (in C#), you have a couple of options.
You can use something like SlimDX to get direct access to DirectX. This provides a clean C# interface to DirectX programming, which lets you target a lower entry point than XNA's default requirements.
You can also do 2D game programming using WPF or GDI+ rendering.
One possibility is the Unity 3D game engine. It runs well, even on low-end systems - as their System Requirements says, "Any graphics card made in this millennium should work".
It uses any of 3 scripting languages, C#, Javascript or Boo (a .Net variant of Python). You can also deploy in either a web-browser or stand-alone executable (Mac or Windows, no Linux yet).
Granted, this adds to the learning curve, as you'd be learning the Unity system as well as C# - but Unity isn't that hard to learn, and it's also a very good game engine. See the Licensing page for a more complete list of features.
Start with trying to make something fun and simple, without worrying about graphics, sound or any of those details. If you like strategic board games, trying to create a computer version of one of your favorites is a good place to start. Making something fun is what games programming is all about and there's plenty of fun to be had with simple concepts and executions.