You can test your connection here: http://ipv6test.google.com/
Apparently, Wednesday, June 8th is "World IPv6 Day" when major websites and CDNs will be sending their content over IPv6. For instance, if you cannot connect to Google on that day, you will need to surf to http://ipv4.google.com to do your searches.
My best guess is that most people will not experience any problems. Almost any computer / router purchased within the last 5 years should either be compatible / or easily upgradable. Most Internet Service Providers (ISPs) *should* already be compatible. In fact, you can obtain detailed information about your internet connection here: http://netalyzr.icsi.berkeley.edu/
Please let me know your thoughts on this. Is the current Internet infrastructure ready for the switch??