(Edit 2010-09-17: So Songbird for Linux has been dropped. See Songbird for Linux dropped, Nightingale picks up. Nightingale has nothing downloadable yet either. Anyway, the point is, there is no point. Rhythmbox Music Player has become good enough now and I've since stopped using Songbird.)
The music player situation on Ubuntu Linux is, unfortunately, a little fragmented and a bit confusing for people who are used to programs like Apple's iTunes.
Of all the choices available, however, Songbird is very promising and is the best I've found for organizing and playing a locally stored library of music files. It has a pleasing UI, familiar to an old iTunes user, although it lacks features for effectively organizing podcasts and streaming music (it plays them fine in the sense that you could manually add them in to play). There are add-ons for streaming music sites like Last.fm, but apparently not for organizing manually added streams.
I hope it'll develop to the point where it'll be like iTunes of, say, around 2008, just before all the iPhone support got stuffed into iTunes (only because the iPhone related features are of indifference to me).
Having said that, there were a number of issues I had to deal with in getting Songbird to work, so here's my laundry list of them and the solutions I found (recorded here for my own benefit):
(1) How to Install. The Ubuntu Community Documentation for Songbird is a good place to start. Bottom line, it's probably easiest to just download it from the Songbird website directly and place it in your own home folder, then add a launcher icon for it (see the Ubuntu documentation).
I've tried the getdeb.net solution before, but it was of no advantage to me on a system for a single user. If you're managing a multi-user system, you should know your own way around packages or putting things into the right bin folder anyway.
(2) Playing streaming music. It was frustrating figuring this one out since the URI I had before worked just fine in Totem. I had to find the .pls URI location to have Songbird play the stream, but I apparently couldn't save it as a "file" to organize along with my other local music files. If you want more information, this and this bug report may be useful, and the support Q&A here looks useful too.
(3) Disappearing playlists. Make a playlist, exit the program and reopen only to find all the playlists missing. I couldn't find anything online about how to fix this, other than some chatter about how it's maybe some add-on's fault. Well, I disabled all my add-ons and the problem didn't go away. Fortunately, since I've just been testing Songbird, I didn't have a large library to care about, which meant I could test my hunch that my Songbird user profile was broken somehow.
My solution? Go to my home directory and delete the .songbird2 folder. Problem solved.
(4) Podcast subscriptions. Super easy to do, apparently also super easy to not work, but definitely super difficult to find since the word "podcast" doesn't appear in the UI (why?). Best description of how to use Songbird to subscribe to podcasts is at the support forum, although the UI for podcast management isn't too advanced yet.
Now if only Songbird could be a better podcast subscription manager, and if only it could organize the streams I manually add, I'd be happy.