There is no tutorial for the API, nor is there likely to be one -- at least officially.  There *is* [**a "Hello World" introduction**][1].

To use the current API (version 2.2), you really only ***need* to know/do**:

 1. How to make an HTTP `GET` request, in the language of your choice.  There are also [various user-supplied libraries in .net, PHP, Python, etc.][2] to help with this.
 2. How to parse the JSON results.
 3. Monitor the results to make sure you haven't busted your quota or received a `backoff` warning.

For the next step, **It's highly recommended** that:

 1. You [get an API key][3].
 2. You learn and use [Custom Filters][4] to streamline operation and save resources and bandwidth.
 3. You know how to [page through results][5].

Optionally, **for sensitive data or write operations**, you need to:

 1. Know how to authenticate in the language and/or library of your choice.
 2. Make an HTTP `POST` to write data.


----------
Beware that **you currently cannot duplicate much of the functionality of Stack Overflow**, using the API.  The (public) API is mostly read-only, and none of the Review or Moderation operations are supported.  
**Update:** Many new write/post methods were added with API version 2.2.  

See [**the API Change Log**][6].
 


  [1]: https://stackapps.com/questions/2/api-hello-world-code
  [2]: https://stackapps.com/questions/tagged/library?sort=votes&pageSize=50
  [3]: https://stackapps.com/apps/oauth/register
  [4]: https://api.stackexchange.com/docs/filters
  [5]: https://api.stackexchange.com/docs/paging
  [6]: https://api.stackexchange.com/docs/change-log