I want to integrate Google Play Game Services for Apple's iOS and for Android. How can I do this efficiently?
You will have to use a plugin to do that. I've used this one and it was straightforward to set up and use (works for both iOS and Android):
Some things to keep in mind (because the debugging log messages shown won't help you understand what really happens, all of them reporting that the application w/ the given key was not found on the server):
When sending your .apk over to Google, enter the SHA-1 from your certificate manually (extract your certificate from the .apk and run the keytool on it). Sometimes Google screws up automatically doing this and the services won't work afterwards.
Be sure to log in with a preset test account when testing (obviously).