I just started my adventure with the Redmine 3.0, the project manager.
I can't Google out if there is an option to export its calendar to e.g. thunderbird-ligthning. There are some plugins, but they are not compatible with current Redmine 3.0. I couldn't find out anything relevant in the changelog, too.
Can anyone tell me from their own experience, if (and how) this feature is available?
Generally, anything that exports to the ics format can be made available to Thunderbird/Lightning, or any other calendaring program. The other option is to make use of the caldav protocol, which is also standardized.
I have heard reports that this fork of the redmime ics export plugin works for 3.x, but I haven't tested it myself. Maybe you can give it a try.
If you would like to create your own, I'd suggest adapting sabreDAV to use redmime as a backend. As an example for how to write a custom sabre frontend for a different service, check out this work.