You have a freewheel and are trying to replace it with a cassette, that will not work. See this article from Sheldon Brown that explains the difference.
http://www.sheldonbrown.com/freewheels.html
You could maybe find a freewheel with 13 teeth on the smallest cog but that is as low as they go due to the way that they are attached.
The LBS was correct, any regearing would cost more than is reasonable to spend on that bike.