Magento – redirect product add to cart button to checkout page instead of cart page

From the comments I conclude that you don't have the soap php extension enabled.
This is not a problem normally for Magento because on the checkout it doesn't require SOAP for anything.
Either you have an extension that does something with SOAP when saving the billing address , maybe a shipping method that retrieves the shipping costs 'live' or the error messages logged in system.log come from somwehere else.
I suggest enabling the soap extension and trying again.

Then config file i have added events observer,

<config>
....
    <frontend>
        <events>
            <controller_action_postdispatch>
                <observers>
                    <controller_action_after>
                        <class>membership/observer</class>
                        <method>membershipControllerActionPostDispatch</method>
                    </controller_action_after>
                </observers>
            </controller_action_postdispatch>
            <add_to_cart_after>
                <observers>
                    <add_to_cart_after>
                        <class>membership/observer</class>
                        <method>membershipCheckout</method>
                    </add_to_cart_after>
                </observers>
            </add_to_cart_after>
        </events>
    </frontend>
.....    
</config>

and My Observer file,

class Att_Membership_Model_Observer
{
    public function membershipControllerActionPostDispatch($observer)
    {
        if($observer->getEvent()->getControllerAction()->getFullActionName() == 'checkout_cart_add')
        {
            Mage::dispatchEvent("add_to_cart_after", array('request' => $observer->getControllerAction()->getRequest()));
        }
    }
   public function membershipCheckout($observer)
   {
      if (!Mage::helper('customer')->isLoggedIn())
      {
         Mage::app()->getFrontController()->getResponse()->setRedirect(Mage::getUrl('checkout/onepage', array('_secure'=>true)));
      }
   }

}

this gives solution for me, the addtocart go straight to checkout page.

Thanks @sander