Before you start, you might want to run down to your local Apple Store and pick up a Mac, if you haven't already got one. Just kidding! The code is the same no matter which platform you choose, but as the iOS one is the hardest with most moving parts, we'll focus on this. In fact, this is a lot cheaper than buying a Mac. There are, in fact, cloud providers that will give you a virtual Mac in the cloud (for instance, the commercial provider: http://www.macincloud.com). The reason you need a Mac is because of the way Xamarin actually gets compiled. When you're writing code in C#, it will not run inside a runtime on iOS, but it will be compiled down to a native language for iOS.

In order for Xamarin to be able to do this, it relies on tools found in Xcode, which is only available on Mac. The same approach is for Android. It does not have a Common Language Runtime (CLR). So, it compiles down to what is right for Android and runs it. Any references you have will be included, but...