If you want to start programming, I'd recommend Lua, Python, and Ruby - in that order.

Scripting languages are very important, especially in games. In fact, video games were one of the first pieces of software to regularly use multiple languages for a single project (for example, an engine written in C or C++ and the game logic itself written in a higher level language like Lua or Python). Virtually every game out there, except for really simplistic ones, will have a scripting system of some kind. Knowing how to use these languages is critical to being able to work in the industry.

From there C# is a great direction to go, partly because of XNA (which will get you comfortable with working with the kind of tools used in the industry), and partly for some diversity in the kinds of projects you work on.

Once you have a solid programming basis and understand the overall concepts, you can move down to lower-level languages like C++. Definitely do not start with C++, though, if you can at all possibly avoid it.

I haven't used Lua or Ruby at all, but Python is very useful if you need to slam some kind of tool or simple program out. Only for heavy duty number crunching is C++ better than Python; for non-game/simulation/3D-modeling applications, Python's ease of use is more important than C++'s speed. Also, most common C++ libraries have Python wrappers these days, and I've heard it's not too hard to create them with Boost::Python.

Once you've learned something like Java or C++, you can usually pick up another language, to the point that you can be productive in it, in under a week. Once you've got the concepts down, it's just a matter of figuring out the syntax, and checking the library documentation when you hit snags.