When you want to surf a new page,  your browser is doing its best to fetch your favourite website as soon as possibile.

Well, this is true, but how can I get the IPv6 version of that site?

Every new browser will ask the OS to get both A and AAAA records on a given url. Then the Happy Eyeball algorithm takes over, and always¹ prefers an IPv6 AAAA record over an A one.

Since 2011, as long as the desired content is v6 ready,  every dual stacked host is transparently moving its traffic on IPv6 without any additional transition mechanism in between.

¹ Nowadays all browsers implement the Happy Eyeball algorithm as per RFC6555 , with the exception of Safari which prioritise the record received the first.