I want to create a Visual Studio 2008 project which contains some example code of how to use an API and then make the project available to download so others can just double-click the project, see the code and run the examples without making any changes.
What files (e.g. .suo, .sln, bin & obj folders?) do I need to include so that people can run the project?
I want to exclude anything that may contain any settings specific to my local machine.
Don't include the .suo file, the .csproj.user file, the bin and obj folders.
This usually matches what you don't include in source control. If that is the case, you can create your distribution by doing an export from your source control system (svn export, git archive, hg archive).
Or write a batch file that executes the necessary commands (zip) to build the distro.
It is easy to test the distro yourself. Just unzip/untar/copy the distro to an empty directory and test there.
This SO question is about maintaining multiple configs.
Exclude .suo, .user and bin and obj folders.
Include .sln, .csproj (assuming C#), any source files (.cs), any resources such as images, and .config files.