API Backwards Compatibility Best Practices

I have an installed Windows client and a hosted service (A) to which it connects. Both client and server and server are developed by my team. And they are deployed. We also have a website that is served by a service (B) that is separate but shares code with service (A). We are now trying to find best practices that will help us ensure backwards compatibility. For example,

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-If we make change service B, it must not change what service A returns such that the Windows client breaks

-If we update the Windows client and/or service A, we cannot guarantee all users will upgrade immediately. Therefore, if service A is at version N+1, it must continue to serve clients at version N. This will hold true for a certain number of previous versions, so N-1, N-2, etc.

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Obviously, testing is huge, but I am wondering what are some other best practices/tools/processes that can help us maintain multiple versions of our software.