Hello,
If I add something into the cart and then click on "proceed to checkout", I'm redirected to the same page that I were and the cart is empty.
The cart module you're using is not the official one and the button "proceed to checkout" is not performing the redirection to the checkout but updating the cart content with incorrect data.
That module is provided by your template and I think that it's not compatible with HikaShop 3.x and newer.
I don't know why your "proceed to checkout" is a submit button while we are using a link (which is displayed as a button, but it's a link).
Right now, I can't see specific issue with HikaShop but I suspect that your template have some customization which are no longer working ; I think something made for the legacy add to cart and legacy checkout.
Regards,
Jerome - Obsidev.com
HikaMarket & HikaSerial developer / HikaShop core dev team.
Also helping the HikaShop support team when having some time or couldn't sleep.
By the way, do not send me private message, use the "contact us" form instead.