I was in about the same situation when our System Admin/DBA left and I just inherited the job. That is a stressful way to start, but I learned fast. I did eventually get some training through an IBM business partner - after I had already done it for awhile. That actually makes sense from the perspective that then you really know where your weaknesses are in your particular situation.
If you are particularly interested only in the DBA role, then learn business object modeling & data modeling and SQL SQL SQL. Learning to design, document, and diagram your database in the context of business objects and their business relationships is important; learning to document the database itself is critical. There are lots of books and web pages available, I think I started with The Handbook of Relational Database Design. For SQL, I think I have SQL for Dummies, SQL for Smarties, SQL/400 Developers Guide, DB2/400, and DB2/400 SQL reference all sitting on my shelf - full of highlights and marked spots. There are useful functions in SQL that just don't seem to turn up in places like MS Access, for example, "Coalesce," which I use often. The web is also good - there are lots of tutorials, answer wikis, blogs and dedicated sites. You will need to work with short scripts at a minimum. Programming is good... Also, try doing the things you do in MS Access with the GUI using the SQL text editor instead. Design one in the GUI and open it in the SQL editor and see if you can improve it...
If you are interested in just the DBA role, you will likely want to investigate other database platforms like SQL Server, other flavors of DB2, and Oracle. If your interest lies primarily with System i, then you should develop some programming skills (CL, Java, RPG...etc) and learn whatever you can about system administration - in the System i environment the DBA role is frequently combined into the system admin role. Don't be reluctant, you can learn that part, and it is fun. I took a 2 day class in CL programming - it is pretty easy and it allows you to automate tasks like crazy.
I hope this is helpful. Best wishes!