How to use HTTP.GET in AngularJS correctly? In specific, for an external API call?

First, your success() handler just returns the data, but that's not returned to the caller of getData() since it's already in a callback. $http is an asynchronous call that returns a $promise, so you have to register a callback for when the data is available.

I'd recommend looking up Promises and the $q library in AngularJS since they're the best way to pass around asynchronous calls between services.

For simplicity, here's your same code re-written with a function callback provided by the calling controller:

var myApp = angular.module('myApp',[]);

myApp.service('dataService', function($http) {
    delete $http.defaults.headers.common['X-Requested-With'];
    this.getData = function(callbackFunc) {
        $http({
            method: 'GET',
            url: 'https://www.example.com/api/v1/page',
            params: 'limit=10, sort_by=created:desc',
            headers: {'Authorization': 'Token token=xxxxYYYYZzzz'}
        }).success(function(data){
            // With the data succesfully returned, call our callback
            callbackFunc(data);
        }).error(function(){
            alert("error");
        });
     }
});

myApp.controller('AngularJSCtrl', function($scope, dataService) {
    $scope.data = null;
    dataService.getData(function(dataResponse) {
        $scope.data = dataResponse;
    });
});

Now, $http actually already returns a $promise, so this can be re-written:

var myApp = angular.module('myApp',[]);

myApp.service('dataService', function($http) {
    delete $http.defaults.headers.common['X-Requested-With'];
    this.getData = function() {
        // $http() returns a $promise that we can add handlers with .then()
        return $http({
            method: 'GET',
            url: 'https://www.example.com/api/v1/page',
            params: 'limit=10, sort_by=created:desc',
            headers: {'Authorization': 'Token token=xxxxYYYYZzzz'}
         });
     }
});

myApp.controller('AngularJSCtrl', function($scope, dataService) {
    $scope.data = null;
    dataService.getData().then(function(dataResponse) {
        $scope.data = dataResponse;
    });
});

Finally, there's better ways to configure the $http service to handle the headers for you using config() to setup the $httpProvider. Checkout the $http documentation for examples.

Using Google Finance as an example to retrieve the ticker's last close price and the updated date & time. You may visit YouTiming.com for the run-time execution.

The service:

MyApp.service('getData', 
  [
    '$http',
    function($http) {

      this.getQuote = function(ticker) {
        var _url = 'https://www.google.com/finance/info?q=' + ticker;
        return $http.get(_url); //Simply return the promise to the caller
      };
    }
  ]
);

The controller:

MyApp.controller('StockREST', 
  [
    '$scope',
    'getData', //<-- the service above
    function($scope, getData) {
      var getQuote = function(symbol) {
        getData.getQuote(symbol)
        .success(function(response, status, headers, config) {
          var _data = response.substring(4, response.length);
          var _json = JSON.parse(_data);
          $scope.stockQuoteData = _json[0];
          // ticker: $scope.stockQuoteData.t
          // last price: $scope.stockQuoteData.l
          // last updated time: $scope.stockQuoteData.ltt, such as "7:59PM EDT"
          // last updated date & time: $scope.stockQuoteData.lt, such as "Sep 29, 7:59PM EDT"
        })
        .error(function(response, status, headers, config) {
          console.log('@@@ Error: in retrieving Google Finance stock quote, ticker = ' + symbol);
        });
      };

      getQuote($scope.ticker.tick.name); //Initialize
      $scope.getQuote = getQuote; //as defined above
    }
  ]
);

The HTML:

<span>{{stockQuoteData.l}}, {{stockQuoteData.lt}}</span>

At the top of YouTiming.com home page, I have placed the notes for how to disable the CORS policy on Chrome and Safari.