Hi,
I have no problem with the add to cart on your website either. Basically, the same as zeljkokiss. The checkout is still working fine.
Once you add a product to the cart, if you click on your checkout link, you will see the products in the cart.
So really, that's a caching issue.
The simplest is to change the "after a product is added to the cart" to "always redirect to the checkout". That way, you'll get the products in the cart module and see the checkout. But you'll have problems with the cart display between different users of your website if it's really a caching issue so it would be best to deactivate all the cache and make sure that you don't have a CDN or the cache plugin or a cache option of your template activated.