If you want to be a Level Designer...start designing levels! This may sound a bit intimidating, but the best way to get started is to start practicing. Sure, there's tons of books out there about Level Design and I recommend reading them. But, if you really want to see what Level Design is about...the only way is to jump in and start doing it.
The way I'd recommend you begin is by using a level editor tool for some recent console games. The map creator that's included with current console games like Lego Indiana Jones 2 or Little Big Planet give you all the tools you need to start crafting levels. The great thing about learning on tools like this is that it's limited to what you can create with a console controller, so they are generally very easy to begin using. Start with something very basic and just begin learning how the tools work. Something I find helpful at times is to literally try to recreate an environment or gameplay sequence from the game. This allows you to focus on learning the execution of the tool set and then you can later expand into creating your own ideas.
Boom Blox without any help!
Once you've spent some time doing this you'll then have a basic understanding of what goes into Level Design. Yes, levels are very different depending on what game you are working with, but the basic principles are the same. Now I suggest moving into a more mainstream level editor like UnrealEd or Hammer. UnrealEd is included with versions of Unreal Tournament or Gears of War while Hammer is included with Half Life 2. These editors are pretty commonly used by game studios, so any experience you have using them will look great on a resume and help you speak intelligently about game development tools. The best interview candidates have working levels to show and if they are in editors like this, people will be able to quickly see examples of what you are capable of.
I do recommend using learning aids for something like UnrealEd since it's a pretty robust tool. I purchased an edition of Unreal Tournament 3 that included a suite of 3D Buzz tutorials (which can be viewed on his site) but I also hear that Mastering Unreal Technology, Volume I: Introduction to Level Design with Unreal Engine 3 is a really good book for getting started.
In upcoming posts, I'll delve into specific issues with Level Design so let me know if there is anything you'd like to see covered.