The IRC channels are the place to go if you want immediate answers to your questions. Be advised that they are a pretty informal place (especially the #musicbrainz channel) and that although they are usually fast, you won't always find someone who can answer your questions.
The mailing lists are discussion centres for different purposes. The most active and important for users are the Users list (for doubts and general discussion) and the Style list (where decisions about most changes to MusicBrainz are taken by the community). There is also a Developers list, for more technical discussion about development.
The MusicBrainz forums are a good place for asking and answering specific questions if you don't want to subscribe to a mailing list. It has two sections, support and discussion, with five forums in total. Note that the style discussion in the forum is not a substitute for the Style mailing list, and while it is sometimes useful for discussion, the decisions are taken in the list following the proposal procedure.
While this wiki is mainly a source of documentation, it can sometimes be used for communication. Every page has its associated discussion page, and you can leave messages for other wiki users in their user talk pages. Be advised though that not everyone checks the discussion pages often, so forums, mailing lists and the IRC are probably better places to go.
The MusicBrainz blog is where big announcements about the MusicBrainz database, Picard and the community are published. While comments are open on it, letting people provide short feedback, it is not really convenient for in-depth discussions, which are usually held in the mailing lists.
You can follow the MusicBrainz twitter (@MusicBrainz) to get some updates about the project. It usually mirrors the contents of the blog, but we will sometimes tweet about smaller issues that are not worthy of a full blog post. You can ask questions to it, but it is not the best place for it, so if you're already here, you could as well use the forums or the IRC!
More or less once a year, the developers and part of the community meet in real life for a MusicBrainz Summit! New summits are usually announced in the blog well in advance.