I don't think there is any such tutorial on APIs. Most external websites (eg paypal) offer their own custom API and instructions on how to use it. This is different for any company that has its own API so you're not going to find a one-size-fits-all tutorial. Most of these companies will offer you an example of how to use their API though.
As for cURL, yes you'll probably need to learn to use it but it really isn't that hard. If you're able to use and understand the php.net manual, then the php.net cURL manual is just as easy to understand. It's about calling a few functions, passing the correct parameters (which php.net has a list of) and then executing the request. There is nothing overly complex about it but if you don't get to grips with it easily, there are always user comments on php.net and plenty of samples on google!
My helpful sig is on vacation trying to loose some weight. It got a bit fat and caused a few problems but it will be back at some point!