> I don't know exactly how the 301 works, and I don't want to lose that optimization for those pages that no longer exist.
If they are gone, they're gone, and so is your optimization. If you mean you don't want to lose the traffic, the PageRank, and the link-popularity, then a 301 will preserve all or most of that.
> One more question assuming a 301 is the way to go, is it feasible to redirect 5 or 6 pages to the index page? My understanding (very limited mind you) is that a 301 says something to the effect of "this page has now moved here". If this is the case then wouldn't it be a mistake to remove the pages that no longer exist since they do no have identical pages in the new site?
You should redirect to new pages that are relevant to the request for the old pages. If there is no feasible direct replacement, then maybe you've got a site map for that subject-section of the site that would be appropriate. If not, then providing some kind of resource that will help the user find what they wanted is the right thing to do.
It's not a very good idea to just outright delete pages that are getting traffic. Providing some kind of logical replacement page is better. If you have no other options, then redirecting to the home page is a last resort, but I prefer returning a short, apologetic explanatory page with a 410 response (use 404 for HTTP/1.0 requests, though) and a text link to my site map and home page. On this page, I also put a 15-to-20-second meta-refresh to the site map.
So the bottom line is to use 301 if you have a logical replacement, and 404/410 if not. This is the protocol defined by the HTTP [w3.org] specifications, and things work best of you follow those.