I essentially did this about 25 years ago. After ~12 years on fretted electric, I bought a double bass, a thick book of Sinatra standards and five or six of his CDs that contain many of the common standards. I then locked myself in the woodshed for about three years, emerging only for the most basic essentials, which included the purchase of several hundred jazz CDs to further my study.
There's no one way to do it, but if you really want to do it, you need to take stock of where you are at:
- How's your fingerboard knowledge? While jazz will help you to expand this, you need more than a basic knowledge of your board to really dig into it.
- How's your theory? Again, while jazz will help you to expand this, you'll likely need more than a basic knowledge of simple diatonic theory. While not an absolute prerequisite, a strong understanding of theory will help you to overcome some of the harmonic challenges you might find along the way. For example, in most sheet music for older jazz standards, the convention used is that of the old Broadway/Tin Pan Alley scores. You'll see these chords such as Ab-6 which won't compute at first. These chords typically mean a iib5-Vb9 of ii. In the case of Ab-6, it is often denoting an F-7(b5) Bb7(b9) resolving to an Ebmin.
- How's your reading skill? While much of jazz is improvised, there are certainly tunes that have a written bass line for at least part of the piece. If you take a big band gig, you might read most or all of the night. You'll also need basic theory and reading skills to understand simple chord symbols and be able to improvise a bass line from them. Understanding how a chord in bar #2 might be harmonically-related (or not) to a chord in bar #5 will help to dictate how that line is created.
I don't mean to dissuade you; rather, I hope to give you an idea of what to expect. This can be an extremely rewarding undertaking, but the reward is commensurate with how much you put into it. Rather than spending money on a teacher at this point, try to dig into it and find your level of interest and, of course, your inevitable confusion. Then, bring questions to a teacher.
Since you mentioned blues, below is a simple example of how jazz might expound on the blues form:
Typical 12-bar blues:
| F7 / / / | ./. | ./. | ./. |
| Bb7 / / / | ./. | F7 / / / | ./. |
| C7 / / / | Bb7 / / / | F7 / / / | ./. |
Typical 12-bar jazz blues:
| F7 / / / | Bb7 / / / | F7 | ./. |
| Bb7 / / / | ./. | F7 | A-7(b5) / D7(b9) / |
| G-7 / / / | C7 / / / | F7 / / / | ./. |