In addition to what a_horse_with_no_name and bhamby said, I find IBM Information Center quite helpful. IBM developerworks can be handy.
I would also recommend watching some episodes of the DB2NightShow. Season 4 should be starting in a few weeks. It is free live web episodes regarding either DB2 LUW (Linux, Unix, Windows) or DB2 for z/OS, where they invite in different speakers to talk about different aspects of DB2 from their real-world experiences. I only attend the LUW ones since that is what I deal with where I work.
Ember Crooks' blog has also been helpful to me.
And if you can nab a copy of Understanding DB2: Learning Visually with Examples, I'd highly recommend it. Expensive, but maybe you can ask your employer to buy a copy of it for the IT department library. I borrowed a copy that was purchased for a co-worker of mine. Unfortunately, the book is still updated only to DB2 9.5 (not 9.7, 9.8, or 10.1), but it still has some really relevant and helpful topics when you are just becoming a DBA. (Same as what happened to me a few months back.)
But yes, most definitely download and install DB2 Express-C, since it is free. And it is the exact same engine as DB2 Enterprise, just with some features turned off. (You can actually turn them on by buying a license from IBM to turn your Express-C into Enterprise Edition). Also to note, IBM has deprecated the clunky Control Center GUI. It is now recommended to download IBM Data Studio full client and IBM Data Studio administration client to interact with DB2. Those are also free tools.
Note that most of the links to documentation is for DB2 10.1. If you have an earlier version of DB2, you may need to change the Information Center URL (modify the vxrx area of the URL).
Welcome aboard as a DBA! I think you will find working with DB2 is fun. I know I do! :)