Okay, so manu of these answers are just plain wrong. In short, you shouldn't care as the biggest impact will be to network admins. They are the ones who have to configure routing and handle everything else that comes with new addresses. The rest of the world simply doesn't know or notice whether they are using IPv4 or v6. Business as usual.

If the question is whether you should play with it at home. Sure thing if you have the desire to. It's the future and only a matter of time before it becomes a reality. Said network admins and ISPs have been delaying the transition since they are the ones who have to work it out and putting your entire user base behind single IPv4 NAT is simpler than moving everything to IPv6.

From network admin perspective, yes it's worth moving to IPv6 since network topology becomes far simpler with it. Fewer sub-networks, and routing rules to handle those. Less hardware to handle NAT and other stuff. Problem is, they made the bed for themselves and switching to IPv6 becomes harder the more you delay it. Number of users in past 10 years or so has skyrocketed. Easily quadrupled. We use to have home computers with dial-up. Easy enough, assign IP when you connect, release it on disconnect. Then broadband came and everyone is sitting online 100% of the time. Then mobile phones which are also online 100% of the time. Then smart devices, now cars and other devices start having public internet access, etc. As number of users increases, network admins keep adding complexity to their networks to handle them. If you don't have public IP, just do traceroute and see how many internal network hops you have.