Note: This tutorial speaks specifically to my observations regarding the described processes. Your experiences and outcomes may vary.
As of July 30, 2013, with a Chromecast dongle and an iPad you will be limited to casting videos to a TV via the NetFlix and YouTube apps. Both apps also worked with my iPhone 4.
At the time this guide was prepared, a web browser was the only way to configure a Chromecast while using an iPad. A Chromecast iOS setup app is expected in the iOS App Store in the future.