At the ISP / hosting provider level most companies have either already upgraded their hardware (and software) or are in the process of doing it.
However there are still quite a few things that don't work properly under IPv6 (or just aren't fully supported / as well supported)
For example, due to the sheer number of IP addresses possible in IPv6 a lot of things simply won't scale. At least not with the current technologies / hardware readily available
Firewalls are another area that is "fun", as getting some of the more advanced things to work under both IPv4 and IPv6 simultaneously is far from easy
And as for your own equipment ..
Most domestic and small business routers, firewalls and other network equipment currently do not support IPv6. Some can be patched / upgraded, but a lot will have to be replaced.
I've been using a pretty nifty device at home for the last few months which does support IPv6 and has a load of other features: FRITZ!Box by AVM
And with regard to websites / dns etc., for some of it you won't have to worry, as it will be handled for you, but some software (MySQL springs to mind) still doesn't support IPv6 properly, so there can and will be issues to overcome.