I would like to have the same content displayed for both domain names, but I would also like to have the Search Engines list both domains. How can I go about doing this with out getting penalized for having duplicate content?

Having two domains showing the same content is redundant, and the search engines consider it spam. Why would a search engine want two of its listings to show the same content? It just clutters up their results and doesn't do searchers any good.

If I just redirect the 2nd site to the first site I don't think it (the 2nd site) will get listed on the Search Engines, will it?

No, it won't, but that's the way to do it if you want to avoid potential trouble (penalities, banning, etc.). Choose one as your main domain and do a 301 redirect on any additional domains so that those URLs all send people to the main URL.

If I want listings for both sites, should I just start from scratch with the 2nd site and create all new content for that site?

Yep. Amen. :) Make sure it's unique content. It's okay to link from one site to the other, but make sure the content is unique and provides a different value for the visitor.