Separating background noise from your main audio is dependent upon having a frequency difference that is wide enough to allow for efficient filtering of the unwanted frequencies. In the case of your lawnmower, it would be easier to filter this from a woman's voice than from a man's, since a man usually speaks in a lower frequency.
Nearly all editing software that has audio effects (including Vegas, PP, etc.), will have a way of filtering unwanted frequencies. Look for "low pass" / "high pass" effects, etc.
Special sound software like Audition will have the most flexibility with editing audio. Check for shareware solutions also, on places like zdnet.com and tucows.com And, as always, Google is a great tool for finding solutions.