Audacity is a great tool for working with Audio, and it's freely available for PC/Mac/Linux. As well as cutting/pasting and multi-tracking audio, it's also great for removing noise from audio files. It can be slow - and uses a lot of disk space - but this is par for the course with audio files. You can download it for free from http://audacity.sourceforge.net.
If you get as far as Producing an Episode, two audacity templates exist for the show which will get you off to a flying start:
These include our theme music, The Unwelcome Guest, appropriately trimmed to match the usual format. Just add content!