Before you start, you will need Python on your computer, but you may not need to download it.
First of all check that you don't already have Python installed by entering python in a command line window. If you see a response from a Python interpreter it will include a version number in its initial display. Generally any recent version will do, as Python makes every attempt to maintain backwards compatibility.
If you need to install Python, you may as well download the most recent stable version. This is the one with the highest number that isn't marked as an alpha or beta release. Please see the Python downloads page for the most up to date versions of Python 2 and Python 3. They are available via the yellow download buttons on that page.
For advice on choosing between Python 2 and Python 3 see Python 2 or 3.
If you're running Windows: the most stable Windows downloads are available from the Python for Windows page.
If you're running Windows XP: a complete guide to installing ActivePython is at Python on XP: 7 Minutes To "Hello World!". ShowMeDo has two videos for downloading, installing and getting started with Python on a Windows XP machine - this series talks you through the Python, ActivePython and SciPy distributions.
If you are using a Mac, see the Python for Mac OS X page. MacOS 10.2 (Jaguar), 10.3 (Panther), 10.4 (Tiger) and 10.5 (Leopard) already include various versions of Python.
For Red Hat, install the python2 and python2-devel packages.
For Debian or Ubuntu, install the python2.x and python2.x-dev packages.
For Gentoo, install the '=python-2.x*' ebuild (you may have to unmask it first).
For other systems, or if you want to install from source, see the general download page.