You might have heard that AIF (Application Integration Framework) is no longer supported in the new Microsoft Dynamics AX. Though the SOAP-based custom services still are there and work as they did in AX 2012. And the services are provided JSON-based, too. Pretty much everything you need to know at the start is available in a great wiki article, the Microsoft Dynamics AX Services Technical Concepts Guide.
The one thing I’d really like to point out is that there are lots of good examples provided by Microsoft and made available on GitHub. You can use them to learn how things work and as a suitable starting point for your implementations. The root address of the examples is Microsoft / Dynamics-AX-Integration.