any language will do and it should work with the local and internet connection.
Something tells me you have not done a degree in computer science...
Ultimatly you would want to use C/C++ as those give you the best control over your program and can interface directly with OpenGL/Direct3D, the two main graphic acceleration standards. However C/C++ is very difficult to code with and platform dependant so not recommened for beginners.
I would advise you use Java as this would allow you to distance yourself from platform coupling and memory management making life a little easier. Many libraries exist to provide graphic acceleration in Java. If you need more fine control over the acceleration you can also use JOGL to interface directly with OpenGL but that is also not recommened for beginners (books teaching OpenGL have pages numbering in the hundreds).
For networking you will want to support IPv4 and 6. IPv4 is what is commonly in use by most people but will start to be phased out over the comming decade due to address space depletion. IPv6 is replacing it and some ISPs already are allocating their users IPv6 addresses so it is advisable to support these as well. IPv6 also removes the problem of NAT resolution since it essentially makes NATs for private use unnescescary.
As for network interaction, that is mostly done using UDP or TCP. UDP is recommened for games which use server for managing game state while clients are streamed a snapshot of the state relevant to them. In these games it does not mater if the odd packet is lost since the server will soon send more up-to-date packets. TCP is recommened for games where every client is synchronized with the game state and so reliable data transfer is required.
Java has all the network and window (the UI widget, not the OS) stuff built into it. The only thing you will have to concern yourself with is a graphic engine library that is hardware accelerated since the built in graphic libraries are too slow for high throughput purposes like games.
Thus all you need to get started...
1. Java 1.7 SDK, the SDK is needed for the java compilers.
2. Eclipse IDE with Java, very useful programming tool that makes life a lot easier.
3. Some graphic engine of your choice that can interface with Java.