In order to decide which database you want to start with,
I'd recommend learning about general RDBMS design first, so you can distinguish and recognize the pros and cons of all of them.
Then look for market share. Recently looking myself, I've noticed that SQL Server is currently very hot, with recruiters asking like 7 out of 10 DBA's if they can handle SQL Server, the 3 left have to know Oracle and preferably Oracle Applications.
At least that's my impression. All the other ones are extremely niche, which may mean you get paid well if you're one of the few being an expert on these.
In terms of actually getting a position as a DBA, that is much harder, because employers want DBAs with massive experience. Newbies are usually not considered, even if certifications show you know your stuff.
The problem is that, as mentioned by the previous post, experience is the most important factor! This is simply because the issues that DBAs come across are rarely learnt from books, even less so than that is the case with other IT fields. Real-world action is what you need.
Therefore, you may want to look for positions within Support functions, infrastructure / application support etc... anything that lets you play around with databases, anything that you can put on a resume as "so many years experience with database so-and-so", and in that position, try and get more responsibilities in actual administration of the system, if you are having limited "user-only" access initially.
Also, get a free version of one of the major providers and install yourself a nice server, maybe even with a website, to experiment on on a multi-user scale, which could count as experience, if you're running an official website with it - just a thought.
So, bottom line... DBAs usually get into that position initially from some other IT role, where they had the opportunity to grow into it, shadowing a senior DBA at first, getting training, taking over minor DBA tasks etc...
Tough world for Newbies, believe me... I tried.
"My brain is just no good at being a relational Database - my relations suck real bad!"