Uploading Site

What is the most SEO-friendly way to upload a fully revised/re-designed website and maintain (or improve) our Google rankings?

SEO-Friendly way?

You will need to set up 301 redirects from ALL of the old pages to which ever new pages that they relate to. There's a lot of different ways to do that so it's up to you how you implement it.

Just remember that no matter what you do or how you do it, you will always take a temporary hit when you make large-scale changes like that. After that it depends on just how good your on-page SEO is as to if your site will go back to the same rankings.