The easiest option is just to use the copy feature from within the tool (go to Administration and there should be Copy links both at the project and the build level). It will make a reasonable copy of the the configuration and allow you to alter for the specific new project.
The configurations are contained in ${HOME}/.BuildServer/config/${config_name}. There you will find a project.xml file. It contains information which is specific to the given project, but it likely has quite a bit of generic information which can be reused for a new project, including some core version control information and general build practices.
If you're projects are very similar (say they all use CVS with a given base root, all use maven to build, all have artifacts with the same pattern) you should be able to create a template file for your projects and then just write a quick script to install a new build each time you start a new project. TeamCity will pick this up and you'll be able to work on the new project.
I usually use this method when starting a new project that works similarly to an existing one.