Personally, I think the best way is private instruction. It's expensive, but if you find a good teacher, it's extremely valuable. Check the local instrument stores' posting boards, they usually have some bass teachers advertising for lessons. Or you can try the internet.
http://www.harmony-central.com is a great site. Also, try out different teachers. If you aren't really comfortable with a teacher, the odds are their are others out there. A lot of teachers will offer the first lesson for a discounted price, some even offer it for free.
Another piece of advice, is not only to learn some of the bass lines from you favorite songs, also learn the melody. Play the horn part on your bass, or the guitar lick. This will help stretch out in your head some ideas that might work.
Most importantly, play play play play play play play. And, practice practice practice practice. The more you play and practice, the more you'll learn, that goes without saying. If you have the right teacher and the right approach to practicing you'll learn quickly. I can't understate the importance of quality practicing. Quality over quantity any day of the week. Have a specific idea of what you want to study, how much time you want to do each different thing, and go at it. You'll start slowly, adding 5ths and octaves. Then you'll add the 3 and 7 of the chord, then more scalar notes, then chromaticism, and so on.
That all will help with melody, and some rhythm as well. Personally, I like to study drum patterns and the piano parts to some of my favorite songs to get a good idea of some fresh things to do rhythmically.
Hope that helps a little.