As you work with Xamarin Studio, you will undoubtedly have questions about the software and cross-platform development in general. However, you are equally likely to find unique shortcuts and efficiency hacks that others haven't thought of. Therefore, I strongly recommend that you begin to familiarize yourself with the various documentation repositories, forums, and blogs that relate to the technology you are going to be working with so you can give and take with the community.
Xamarin Developer Center is your launching point to documentation, code examples, and training videos provided by the Xamarin team (http://docs.xamarin.com).
Xamarin Forums are an excellent place to start. You can sign up using the same login you used to download Xamarin Studio. Take time on a regular basis to read through the posts. You will be amazed by the development gems you will pick up, even from casually reviewing the topics (http://forums.xamarin.com).
Xamarin's Bugzilla server is an important site to become familiar with. If you find bugs in the software, this site will help you inform the community. Likewise, you can research a bug to see if someone else has already posted it; if so, see whether a workaround exists until a fix can be pushed out (https://bugzilla.xamarin.com).
In addition to the Apple Developer Program, you should familiarize yourself with the Apple Developer Library, which contains documentation on developing all things on Apple (https://developer.apple.com/library/).
The iTunes University offers many resources on iOS and mobile development completely free of charge (https://www.apple.com/apps/itunes-u/).
Android Developer Library contains a wealth of information on mobile development. If you are new to Android development, take the time to review the App Fundamentals tutorial at https://developer.android.com/guide/index.html.
GenyMotion provides some of the fastest Android emulators for app testing and presentation (http://www.genymotion.com/).