I just became the webmaster for a youth sports organization. I looked over the site that's uploaded and it's a mess. Duplicate pages, no index page, etc. I would like to just delete the whole thing and upload the new site I just designed. What's the best way to go about cleaning up the old site?

Make sure you redirect all of the old pages to the most relevant page in the new site.

It depends, if your site has a good established authority, I wouldn't just delete the whole thing.

So, before redoing everything, you should think at :
Google PR of the site, position in SERPS for different keywords.

A good tip would be to install Google Analytics script and monitor for a few days the top visited/landing pages. Then, definetly do a 301 redirect on those pages.