To get the transaction details from an omnipay response, there are a few methods you can use on the response object.
// how to grab the transaction date, id other useful information to store.
if ($response->isSuccessful()) {
// the transaction date is right now
$date = gmdate();
// the id
$id = $response->getTransactionReference();
// other useful information (the raw response from Stripe)
$data = $response->getData();
}