Most of the resources mentioned here are a bit advanced , used for 3D games, and requires quite a bit of reading to actually use in a game.

Here are a few libraries that should hopefully help in making simple games, with a much lower learning curve.

What do games need?
Graphics -> something draw on the screen, primitives (lines,circles,quads) and textures (pictures)
Sound -> bangs and bams, background music to make the game come alive.
Input -> To let the user actually interact with the game

So here are some simple libs which can help you with the above:

Frame Buffer: http://www.pixeltoaster.com/
Something to draw on. For beginners, it's recommended you use fairly low-level APIs and do things yourself. This should help you get a buffer to draw on and to learn some rendering techniques (and to develop a greater appreciation for the reasons why somethings are the way they are)

Rendering: http://freeglut.sourceforge.net/
Once you've learned to draw, you can let an API do it for you. Freeglut is pretty hand for that and can do wonders to simplify OpenGL initialization and rendering across various platforms.

Image Loader: http://corona.sourceforge.net/
Every game needs it's art. This lib makes loading images in a variety of formats easy. It can be used with any application. It just loads the image from file into a buffer in any mode(eg RGB).

Sound: http://audiere.sourceforge.net/
Open Source sound library. Like FMod, it can load and play a variety of sound formats like .wav, .mp3, .ogg, etc. It is completely free and very very easy to use.

Input: O.o
Sry, this is still platform specific. BUT ... PixelToaster, Freeglut etc have their own simple methods for getting user input.


Compiler: http://www.codeblocks.org/
A very nice and simple compiler IDE to use. It has many similarities with the Visual Studio C++ IDE and is extremely easy to use. A huge plus for beginners is that it has support for DevPaks, which are used to download and setup libraries easily. It means you just have to open up a little tool box, select the libs you want, hit download, wait for them to finish and then be automatically setup for use. You can also get project templates for these so that all paths and libs are already included for you to start coding.


Oh and all the above mentioned tools are cross-platform, so it does not matter if you are on windows or linux (or mac).


And to knock you out:

Scripting: http://www.somedude.net/gamemonkey/
Poweful C++ class based scripting API. Some people cant stand anything other than LUA but GameMonkey is quite easy to use, smaller and has an encapsulated interface. Not as powerful as LUA nor as much documentation, but still wonderful for games.
Others: LUA
Squirrel

Physics 2D: http://box2d.org/
An adorable 2D physics library. Simple and cross-platform (with bindings for other languages). It's really easy to learn and add physics to your games.


That should give you all the basic APIs to develop any game you can think of in 2D. I've not mentioned any other libraries for particles, models, shaders etc all of which can be used in 2D and 3D, but they are out there and you will run into them once you start making your own games.

Enjoy the journey (one from which you may never be able to depart, should you be able to start).