If you migrate to Joomla! then keep in mind that you should use a site template that conforms to XHTML and CSS standards. There are many templates out there that still use depreciated markup and will hinder your SEO efforts.
Changing site software will always result in an "adjustment" period as far as search engines go. Be prepared for that. As old site content drops, new site content will be added. I've never used redirects, so cannot comment if this will help you or not. Others may comment on this after me, however.
There are also components for enhancing Joomla! SEO. There are a few out there, so it will be up to you which one you use. Depending on the server you are running on (supports mod_rewrite?) Joomla! will natively use SE friendly URLs. You just have to set the option in admin configuration and define the .htaccess file.
The best thing to do with Joomla! is to treat your front page as a blog. Constantly update with new articles or content, and SE's will love you.

Installing a SE sitemap may help too.
As mentioned by sleepyhead above, there is a learning curve to Joomla, but most answers to any questions you have can be found in the Joomla! support forums.
Joomla! is a robust CMS and can pretty much accomplish anything you may want to do. There is an endless supply of plugins to enhance it's functionality.
Hope this helps.
An afterthought: You can always leave your static pages in tact then display them using the "wrapper" option in Joomla! These pages will be displayed within Joomla! and will appear to be a part of the new site while leaving the SE positions in tact. Like I said, this is just a thought.