Python can be built from source, but the easiest way is to get a binary package for your type of system and version of Python. This document will give you information on either type of installation.Windows Binary Installer
This is probably the most popular method of installation. If you are running on windows, it is highly recommended you use this form of installing. The installers come with with nearly everything you need, and have an easy point and click installers.
The first thing you will need is an installation of Python. Python binary installers make it easy to get this done. Pygame binaries usually come for the latest 2 releases of Python, so you'll want to be fairly up to date.Once that is in place, you want to download the appropriate windows binary. From the pygame downloads page you can find the .EXE file you need. This will automatically install all of pygame and all the SDL dependencies. The windows binaries have filenames like this; "http://www3.telus.net/len_l/pygame-1.8.0release.win32-py2.5.msi". This would be the installer for pygame version 1.8.0, for Python version 2.5. You shouldn't have trouble finding the correct binary from the "Windows" section of the download page. http://www.pygame.org/download.shtml.
You will also probably want to install the windows documentation and installation package. This will add easy links to the different documentation and games that come with pygame. The installer for this is found next to the other windows binary downloads. The filename looks like this; "pygame-docs-1.8.0.exe". And this would install the documentation and examples for pygame-1.8.0
One other thing the windows binaries are missing is the Numeric or numpy Python packages. You can easily install this separately and it will allow you to use the pygame "surfarray" module. This module is optional, so there is no need to do this. There are binary installers from the Numeric download page. http://sourceforge.net/project/showfiles.php?group_id=1369. A Numeric for Windows python 2.5 can be found on the download page: http://www.pygame.org/download.shtml.
Numpy is newer than Numeric, so you should probably use that... however both are not entirely compatible. Instead of numpy you can also use PixelArray, which is built into pygame.
For many unix systems, the easiest way to install pygame is from source. Still, there are binary packages available for different distributions.There are several binary RPM packages for linux machines. These are actually a little bit of work to install, since you will also need several RPM packages for the dependencies. There is a good chance your linux distribution came with the needed dependencies (like Python and SDL). There are binary RPMs available from the website for each dependency.
For debian systems, pygame is actively maintained in the debian archives. Visit the debian pygame page for more information. http://packages.qa.debian.org/p/pygame.html
FreeBSD also has an active pygame package. While techinicaly it isn't binary, it is automatically built and installed by the ports manager. See the FreeBSD package page for more information. http://www.freebsdports.info/ports/devel/py-game.html
Gentoo has a builtin package for pygame. This is compiled for your system as it installs, similar to BSD, http://packages.gentoo.org/package/dev-python/pygame
For Mac OS X 10.3 and above, binary packages are available from pythonmac.org packages: http://www.pygame.org/download.shtml
This package includes almost of the dependencies required for pygame (SDL, SDL_image, etc.), but you need PyObjC 1.2 or later, and may also want to get Numeric, numpy and PyOpenGL.
To build self-contained pygame applications, you should use py2app. There is an example in: /Developer/Python/pygame/Examples/macosx/aliens_app_example
Compiling and installing pygame is handled by Python's distutils. Pygame also comes with some scripts to automatically configure the flags needed to build pygame. Use the "setup.py" script to start the installation.The first time you run the setup script, it will call the "config.py" script. This will build a "Setup" file which stores all the information needed to compile. The "config.py" will do a good job of detecting what dependencies are available and where they are located. If it isn't perfect, it is easy to build your own, or edit the created "Setup" text file. This "Setup" file is a simple Makefile-like text file. It defines variables needed to use each dependency, and then enables all the pygame modules with found dependencies. If you have trouble compiling, you should be able to easily fix any problems inside the "Setup" file. Running the "setup.py" script will call distutils to build and install the pygame package. Distutils actually supports a wide variety of compile and install options. running "python setup.py help" will start to show you the different options available. You can change many things like install locations, compiler to use, and more. Calling the "setup.py" script with no arguments and it will just ask you if you want the default flags needed to compile and install.
You can compile pygame on windows with mingw (gcc for windows) and also with visual studio. Up to date details can be found here: http://pygame.org/wiki/CompileWindows
Compiling from linux shouldn't give you any problems. One thing you must keep in mind is that most linux RPM packages separate the actual library from the "dev" files needed to compile. To build you will need to make sure the packages like "SDL-dev" are installed.You can check to see if SDL is ready to be built from by running the command sdl-config and seeing if it is found. If the sdl-config script is not on the path (or you have more than one?) Set the environment variable SDL_CONFIG to its location.
Sometimes you will have the SDL libraries installed in once location, and the other SDL libraries in another. This tricks the pygame config scripts, but you can help it out by setting the environment LOCALBASE to a path prefix where the other libraries are. The common case for this is SDL installed in /usr and other SDL libs installed in /usr/local. The command for this situation is "LOCALBASE=/usr/local python setup.py install".
Up to date instructions for compiling on Mac OS X can be found here: http://pygame.org/wiki/MacCompile