What do you need to start typing in LaTeX

On Windows:

First, install MiKTeX system - this is the best choice as far as I know.

1. Download MiKTeX from miktex.org. I installed the basic version. It's very easy to install other packages after the installation.
2. Update your installation using the Updater.
3. From the Settings window, under the Packages tab you can now add other packages from an internet/local repository. A proxy server can be configured.

Then choose an editor. The best programs are:
LEd (www.latexeditor.org)
TeXnicCenter (texniccenter.org)
Texmaker (www.xm1math.net/texmaker)


On Linux:

The best editors for Linux:
Kile (kile.sourceforge.net)
Texmaker (www.xm1math.net/texmaker)
LyX (www.lyx.org)

A nice overview of these editors can be found here.