This page from the IPv6 Day event has quite a lot of information and references to address this question.
I think most relevantly, they state:
What if I have a problem connecting to a participating web service. What can I do?
It's very unlikely you will be
impacted by IPv6 Day. Current
estimates are that 0.05% of users may
experience connectivity issues, and
participating organizations will be
working together with operating system
manufacturers, home router vendors and
ISPs to minimize the number of users
affected. You can test your Internet
connection ahead of IPv6 Day here. In
the unlikely event you have problems
on IPv6 Day, the best thing to do is
to contact your ISP for support. In
the coming months, participating
organizations will be working together
to publish help guides with more
specific instructions for diagnosing
and addressing potential issues.
I recommend you start by trying to factor out your home network from the equation. I would plug a computer directly in to your modem from your ISP and see if you can communicate with that IPv6 website. Hopefully you don't have one of those jerky ISPs that makes you register your MAC address with them, as that'll slow ya down.
From what the reference I've given already says, if your OS is totally up to date, and you can't hit IPv6 stuff through your connection, then you may have an ISP that's not ready for IPv6 yet. I'd call their tech support and ask them if/when they will support IPv6.
Once you've got those two ends going, you can focus on your home network. That gets in to the domain of whatever your home equipment is. You may have to perform firmware updates to get everything IPv6 compliant. In my case, my router had a check box I had to enable to "permit IPv6" traffic. I'd like to think it's unlikely your hardware doesn't support it, since IPv6 has been talked about for well over 10 years now.. But maybe you'll need to buy a new router or something..
Anyway, hope that helps get ya on the right track.