I'm only a beginner myself, but I've been very happy so far with The Hal Leonard Bass Method, $15 for the complete version with cds on Amazon. Back in elementary and middle school I played clarinet for about four years and it reminds me a lot of how I was learning back then except, well, I've probably learned more about bass in a month than I ever managed to learn in four years, though most of that is teaching myself theory.
The thing I like best about this particular book is how it starts with standard notation and gradually introduces the fretboard. Most early lessons involve introducing a few notes on a single string, generally alternating between one lesson on naturals and one on accidentals, learning their position on the staff and fretboard and then playing through a few exercises to get them down and a few simple songs. Rather than feeling overwhelmed at first or just learning notes as needed to play a particular song, you start slow and gain solid knowledge of exactly what you're playing.
Criticizing it is the fact that it can be a bit sparse. Yes, you have a lot of stuff to play, but you don't always get a lot of text explaining the background and theory. Likely this is just to make things a bit simpler for the student. For example, early on it introduces a simple 12 bar blues. While at the same time it explains how the bass line corresponds to chord changes, playing the root of a chord, and introduces the concept of playing triads it doesn't delve much into any of these. I'd say it should at least mention the idea of a I-IV-V progression or provide a side-bar or cross-reference for more detailed information.
In short, it's a good book to start learning and it'll teach you the basics of playing, but this comes at the detriment of learning theory and understanding why you're playing what you're playing.
Many, many people have said excellent things about Bass Guitar for Dummies and I can see how this might be a very good complement to fill in these gaps as it's largely the inverse: far more focused on theory, concepts, and styles than teaching you to play. At the same time, while the information is excellent and well-presented I find the Dummies style to be rather grating and if you're dedicated and search around the Internet you'll probably find a lot of the same information, just not presented and collated quite as well.
