php design question

Why not extend your object instead of including something. APIConnect should be the abstract class and API something should extend it.

abstract class ApiConnect {
    protected $_something;
    protected $_something2;
    protected $_curlResource;
    function __construct($param1){
        /* do curl setup here and save it to $this->_curlResource */
    }
}

class ApiSomething extends ApiConnect {
    protected $_paramForBase;
    function __construct(){
        super($this->_paramForBase);
        echo "Var from base: ".parent::_something;
        /* do more things with curl here, through parent::_curlResource */
    }

    function setParamsForCurlCall(){
        /* add curl parameters with curl here, through parent::_curlResource */
    }

    function execCurl(){
        /* do your final curl call here */
    }
}

Then you can do whatever you want with your abstract class, and you can API something can have all the control logic for the base class. Granted you would only want to do this if your going to extend ApiConnect in more than one class.

First of all I think that Apiconnect should be a singleton (you need just one authentication) and you should rename it to something like APIConnection if you want to use it as a resource in other classes. Extending it would also be an option, but it depends on what your other class does (you other class should also be a singleton if you choose to extend). Making it abstract or not depends only if it can be used as is.

Inside your APIConnection (I prefer naming it like this) you should have a call method doPost or something of the sort that makes a request to the API (keep it as general as possible). In the constructor call the login API method if you want to authenticate when instantiating. When you need to call other API methods do that directly through the doPost you just created.

You can improve your design by following these rules:

-don't do real work in the constructor (see APIConnect)
-favor composition
-inject the dependencies in the constructor
-meaningful class/interface names
-single responsibility

I would refactor this more like this:

interface APIConnection { // responsible for the connection
    public function connect();
}
class FacebookConnection implements APIConnection {
    public function __construct($url, $name, $pass, $params) {
         // set only connection parameters here. NO WORK DONE, NO CONNECTING
    }
    public function connect() {
         // curl stuff
    }
}
class FacebookFarmer {
    public function __construct(APIConnection $connection) {} // inject dependency, composition
    public function harvest() {} // responsible for the actions having the connection
    public function milkCows() {}
}
class FacebookFarmController { // translates what user requests to do to an action
    public function __construct(FacebookFarmer $farmer) {} // injection again, composition
    public function doAction($request) {
        // eg $action = $request['action'];
        // then $this->famer->$action() if exists
    }
}

I don't know whether this is an example of something you could use because APIThis and APIThat is a bit difficult to understand. If I misunderstood please clarify your intentions.