AdrianC

Unity has a free version, why don't you give that a try? For networking, there are many third party solutions including Photon, Smartfox Server and uLink. As an indie, Unity is  by far your best option.

One thing I want to say is, if your goal is to make a game, going the OpenGL is going to waste a ton of time. I mean years, if you want to have anything half decent. If you want to do it as a learning experience, then yeah, definitely go for it, but if you mainly want to actually create a game, then use an engine.

Other then Shiva, I don't know of any engine that you can use and code in Lua (for mobile). If I was you I'd also take a look at Unity before picking an engine, even though it doesn't use Lua. You can code in either C#, Unityscript or Boo.

Umm, that's probably one of the most useful features in the IDE. Ok maybe not, but its super useful.

Well there's quite a lot of things to say about this, but I'll give a quick intro.

First, you have to chose a language to program in. There are many, including Java, C++, C#, Python, etc etc etc. Each language may be better at doing some things and worst at doing others. Generally, the ones I listed are pretty good all around languages. My recommendation? C#.

Next up, you need to get an IDE aka Integrated Development Environment. Inside the IDE you write your code, you can compile your code into executable files, and it provides various tools that help you program. Alternatively, you could just code in Notepad, and use a compiler. Compilers take your code, process it, and turn it into something that your computer can understand. They produce executables. 

Generally, you're better off using an IDE, as it provides this feature, and many more. For C++ and C#, the industry standard is Microsoft Visual Studio. Visual Studio Express is free to use for whatever you need (including commercial projects), however it's missing a few features. As a beginner, you will not need any of the missing features.

Alright, so you downloaded the IDE. Now you will code here for a while. You will create console applications. These are entirely text based applications, that get you used to programming. When starting, you'll probably code a basic calculator, maybe a text based game, or even tic tac toe. 

Once you got comfortable enough with console applications (You've went through a few books, done many practice exercises), you'll want to move into graphics. There are many options here. 

You could code directly in DirectX or OpenGL. These are very low level, and not recommended for actually producing games with, though they make great learning exercises. 

You could grab a framework, or graphics library, that just makes displaying graphics on screen a lot easier. For C#, XNA is great. C++ has SFML, and Python has Pygame. I would stay away from Java.

Or, you could grab a game engine like Unity or UDK. Game engines provide a lot of tools right out of the box, for example physics engine, particle engine, post processing effects, etc. Its generally recommended for beginners to start off with something like XNA, and then move on to using a game engine, as you'll learn a lot more and your code will be better.