Where do checked items go?
 

Just started using OmniFocus. When I've completed a task/item, I check the checkbox next to it and the item will show a strike-through, then upon screen refresh or synch the item dissappears. Where can I find this list of done items?

They are still there, but hidden from view by your view bar settings. One easy way to get them is to use the Completed Items perspective, under the Perspectives menu.

Be sure to watch the introductory videos on the [URL="http://www.omnigroup.com/applications/omnifocus"]OmniFocus product page[/URL] -- both the more in-depth ScreenCasts Online treatments and the introductory videos (including one on perspectives) down at the bottom of the page.