Magento 2: How to send data using Ajax form in custom form?

You can just set below code in your phtml file to use ajax, You have to change your customurl in below code,

<script type="text/javascript">
    require(["jquery"],function($) {
        $(document).ready(function() {
            var customurl = "<?php echo $this->getUrl().'frontname/index/index'?>";
            $.ajax({
                url: customurl,
                type: 'POST',
                dataType: 'json',
                data: {
                    customdata1: 'test1',
                    customdata2: 'test2',
                },
            complete: function(response) {             
                country = response.responseJSON.default_country;
                state = response.responseJSON.state;         
                console.log(state+' '+country);   
                },
                error: function (xhr, status, errorThrown) {
                    console.log('Error happens. Try again.');
                }
            });
        });
    });
</script>

inside your controller file execute() method,

<?php
 use Magento\Framework\Controller\ResultFactory;
 public function execute()
    {
        $resultPage = $this->resultFactory->create(ResultFactory::TYPE_PAGE);

        $response = $this->resultFactory->create(ResultFactory::TYPE_RAW);
        $response->setHeader('Content-type', 'text/plain');
        $country = 'india';
        $state = 'gujarat';
        $response->setContents(
            $this->_jsonHelper->jsonEncode(
                [
                    'default_country' => $country,
                    'state' => $state,
                ]
            )
        );
        return $response;
    } 

Accepted answer is good, but i think could be useful take advantage of the js validation that magento core offers. So, try to use below js script:

<script type="text/javascript">
require([
    "jquery",
    "mage/mage"
],function($) {
    $(document).ready(function() {
        $('#form_id').mage(
            'validation',
            { 
                submitHandler: function(form) {
                    $.ajax({
                        url: "url to module/controller/action",
                        data: $('#form_id').serialize(),
                        type: 'POST',
                        dataType: 'json',
                        beforeSend: function() {
                            // show some loading icon
                        },
                        success: function(data, status, xhr) {
                            // data contains your controller response
                        },
                        error: function (xhr, status, errorThrown) {
                            console.log('Error happens. Try again.');
                            console.log(errorThrown);
                        }
                    });
                }
            }
        );
    });
});
</script>

Don't forget that controller has to return JSON response like:

$response = $this->resultFactory
    ->create(\Magento\Framework\Controller\ResultFactory::TYPE_JSON)
    ->setData([
        'status'  => "ok",
        'message' => "form submitted correctly"
    ]);

return $response;