When the Internet Assigned Numbers Authority issued the last blocks of IPv4 addresses to the Regional Internet Registrars, it wasn't the end of the world, or even the end of the Internet. In fact, it's not even the end of the world for IPv4 addresses. Chances are, you'll still be able to use IPv4 addresses for a long time, probably for years.
What's actually exhausted is the supply of new IPv4 addresses that the IANA can issue to the five regional Internet registries around the world. Those regions can still issue addresses to organizations that need them, and will continue to do so for the next few months.
In fact, the vast number of users of all types on the Internet will likely not notice any changes in their addressing. The major Internet providers all have IPv4 address pools that they can issue as needed and most of them are using private addressing pools and may never need to change. But there are some networks that will see the effects eventually. They include large businesses and other major organizations that act as their own ISPs and have their own pool of addresses. When those run out, they'll need to find a way to react.
But right now, there's no reason to rush madly into IPv6. For one thing, there's virtually no place on the public Internet that is using IPv6, so you will have few choices of destinations. One estimate I saw recently puts the number of IPv6 sites on the global Internet as less than two-tenths of a percent. While there are a few Websites that do allow connectivity using IPv6, those also allow IPv4 connections.
So what's all the fuss about? Eventually, large organizations are going to burn through their pools of IPv4 addresses and will need to start moving to private networks that use NAT (network address translation) or they'll need to start using IPv6 where they can. As demand for IPv4 addresses continues to grow and supply dwindles, IPv6 networking will become more useful. Likely, businesses will find that some types of communications-such as between data centers, or between data centers and cloud providers or even within the data center-are able to use IPv6 with little or no disruption to other activities.
A move to IPv6 will free up IPv4 addresses for end users, for devices that can't use IPv6 and for network infrastructure that isn't or can't be enabled for IPv6. Eventually, portions of the public Internet will start supporting IPv6, but it's not going to happen overnight. And even when it does start to happen, the change won't be all at once.