I just got to integrate all three. A bit fiddly but worth doing.
First of all, Lightning adds calendar and task list functions to your email client.
That didn't want to play the first time I tried adding a couple of tasks, but it cleared itself after a couple of restarts of Thunderbird. Apparently this has happened to other people too.
You switch between email client, calendar and tasks by using the icons at the bottom left of the screen.
So I now have lots of tasks up, some repeating, and two calendars, one for me and one for HPI. Setting up a new calendar is dead easy. In Calendar view go to "Calendar" menu item, then "New", then follow the wizard.
Then I wanted to integrate my Google calendar. This doesn't work with the normal calendar wizard, though it does do other networked calendars. For google you need the "Provider for Google Calendar" which I duly downloaded and installed. But it wouldn't work.
After quite a bit of fiddling about on the forums etc, I deduced that the Provider (version 0.4) was built to work with Lightning 0.8, and I was on Lightning 0.9. I tried downloading and installing the 0.8 version of Lightning, but it wouldn't let me because it knew it already had a different version installed. And it was still obstinate when I uninstalled Lightning 0.9. I guessed that was because it was keeping the data stored somewhere, even after the uninstall, and didn't want to overwrite it. I considered deleting the data, but then decided to try a nightly build on the Provider. I didn't try this before because I'm not a debugger. (Just a bugger, some say.)
Anyway I went off to find the nightly build of Provider here, uninstalled it, installed the new one, and lo and behold it worked. Google Calendar appeared as an option in my New Calendar wizard which-version-do-you-want-page.
There were still a couple of trickinesses to work out, but I was greatly helped by this tutorial at bfish. Note that the tutorial itself is based on old versions. In particular, you don't find "Manage" on Google calendar - it's "Settings".
I found quite a lot of other bits and pieces while I was tracking this down. Not least the developer of Provider says that Google are developing CalDAV for their calendar, a standard that Lightning supports, so Provider may not be needed long term. But he's still supporting it.
And OpenOffice are developing a Personal Information Manager around Lightning.
I'm off to play with Lightning again. At some point I expect the integration of Lightning and Provider to fall over again because the builds won't match, But till then, I'm very happy.
Update 22nd Sept:
Doing this on Tbird in the office today, I found I needed the nightly build of Lightning to work with the nightly build of Provider. Seems like you have to pick your builds.