If you downloaded the DevPak, then use Dev-C++ to install it.

A better idea however is to stop using Dev-C++ as it's outdated, featureless and full of bugs. When you've decided on a better IDE (I recommend Code::Blocks or Visual Studio Express), come back and ask the question again.

Also, that version of Allegro is also old. There's a newer version of the Allegro 4 branch available, and Allegro 5 is also now available.

Now you have two options :
1)
Download binaries of Allegro 4.2.3 or Allegro 5

2)
Download source code for Allegro 4.4.1.1 or Allegro 5.0.0 and build it yourself.

The only way to get Allegro 4.4 built is to do it yourself, as no one provided binaries for it.

If you're new to programming and building libraries, go with option 1, unless you want the latest version of Allegro 4.

To use binaries, copy the contents of the allegro\lib folder to your compiler's lib folder, and copy the contents of the allegro\include folder to your compiler's include folder. Copy the contents of allegro\bin (the dll's) to your progam's folder or to Windows\system32 to install them on your computer.

Once you have allegro installed, your program will need to link against the allegro libraries (*.lib files). Read your IDE's manual to see how to do this.

As a beginner, would it be better for me to download Allegro 4.2.3 or 5?

I think that I have everything downloaded and installed, but I am having some major trouble with the linking. I found some help here,
but it is not helping much. Again I am new to all of this and I am sorry if I might seem naive to anyone.