How to become a DBA...the non-standard route (not beginning as sysadmin or dev)? Background:

I'm a recent BS graduate who's found myself in an extraordinary situation. I've fallen into something of an entry-level Oracle DBA role straight out of the college box. At my small company where the only 2 database people around have each been at it for over half the time I've been breathing, it's difficult to see the path between where I am and where they are. 

From what I've read, many successful DBAs emerge from systems admin or developer roles, and I feel as though I've cheated myself - I've jumped the gun and now have no base in which to root new knowledge. Furthermore, I'm pulled in too many different directions as far as what to learn first. (Must understand B before learning A; but must have a grasp of A before diving into B!)

I often find myself wondering how to get where I want to go. What plan of attack might be the simplest for a beginner to begin? Width before depth or depth before width (i.e. an overview of the entire gamut while glossing over details OR an in-depth dive into RMAN or another specific topic). Then what about how Oracle plays with others like Linux, ASM/disk arrays, hardware, business applications, etc.? 

Endless thoughts! Stressful implications!

Question: 

Essentially, what resources (free or paid) would provide the most value up front, considering virtually no experience and only vague textbook education? Something that may provide exercises, lectures, etc. An ideal source would include more advanced or applied topics than basic SQL and db architecture without too much jargon.
You can add a lot of value to your career at little or no cost except for some of your  time.  You need a desktop computer or server with at least 4 GB of ram. Download Oracle virtual box or pick a pre configured image here.

Install and configure.I learned more by breaking things than by reading in books or videos so don't be afraid to break your database repeatedly as you figure how to use the tools of the trade. With virtual machines you can take a snapshot, break the image and revert to the saved snapshot.

Once you have installed the database you can try adding a schema of your design or play with Oracle APEX to get a quick idea about web applications. 

All the Fusion applications are available for download like BEA weblogic and you can play with some Hello world apps in Java.

Then try and duplicate some of the things you need to do at work like backup, tuning, RMAN and anything else you need to practice on.  It won't be easy but it will reward you with hands on experience which will give you confidence for work tasks.