Whether you want to seriously play jazz bass or not, I think studying jazz is one of the best ways to learn about music and how to play the bass. Jazz requires you to learn about all the elements of music more thoroughly than any other style of music. Since its focus is on improvisation, you're forced to have all the essential bass playing skills right at your fingertips. You have to have solid rhythmic skills, be able to groove, know how to read music, understand harmony and melody, and really know the fretboard well. Even a very basic understanding of jazz bass can make learning most other styles easier.
The first thing we'll learn is how to play walking basslines. We'll start with walking on simple chord progressions like the Blues and some easy jazz standards. Then we'll move on to Rhythm Changes (a common jazz song form) and more advanced jazz standards.
We'll study jazz harmony. You need to know the difference between a C chord and a C7b9#11b13 chord. I'll explain all the common chord progressions used in jazz and how to play on them.
Then we'll venture into jazz soloing on the bass. We'll examine the elements of melody and melodic structure. We'll learn solos by great jazz players (not only bass players, but other instruments too). We'll learn how to play the changes and structure a solo.
I really hope this is a direction you want to pursue. Jazz can seem very scary. It's easy to get lost and frustrated with it. I think one of my top skills as a teacher is to make jazz theory very easy to understand and usable. You'll really get a lot out of it.
Bass Teacher - Tempo School of Music, Houston, TX