I don't have any links like the answer above me, however, I've been playing bass for ~8 years, and taught myself from nothing.
The best advice I can possibly give is to work on your timing, and remember that you're always learning, even when you're not playing. Bass is ALL about the timing. It doesn't matter if you're playing, or you just have an idle hand at work, tap out a rhythm with your index and middle fingers, alternating between the two.
ALWAYS work all four fretting fingers! Especially in the early stages of your development you want a level ground to start from, so you don't want one weak finger in the mix. Same goes for your picking hand, if you intend to use a pick, practice with one. If you intend to play fingerstyle, or slap, practice doing so. It may sound like poor advice, but you'd be surprised how easy it is to fall into a rut of saying 'I didn't have any luck with that last time... I'll just play this.' to yourself if your progress levels off for a while, which it will at one point or another, but don't worry...
Lack of apparent progress does not mean lack of absolute progress. What I mean is that you may not see how you've improved, but a listener who heard you play a month ago will be able to tell the difference a month later where you might not be able to. If you feel like your progress is stagnating, try playing in a different room if possible, or change your surroundings. Sometimes the switch of scenery will inspire a new riff, or new set of riffs to explore. You can also try jamming with some other musicians in your area. Sometimes seeing someone else play can get your creative juices flowing.
Once you have a confident sense of timing and finger control, you can really start to get the 'feel' for blues, and the technicality of jazz. As far as funk goes, I don't have as much experience other than learning how to slap, but again, timing is key.
Blues bass lines have a lot of smooth transitions, (slides and soft hammer-ons, etc...) where jazz lines tend to be a bit less minor, and use hard transitions(A hard hammer-on, or aggressive string strike) more often to accentuate timing and intricacy between bandmates.
Funk has a whole different set of loose guidelines, but from what I've found, it makes use of a lot of hard transitions and odd timing between notes, where blues feels 'late' from the beat, funk often feels more like its anticipating the coming beat.
A metronome is an excellent start, as it gives you a fairly solid sense of timing, but honestly the best way to learn to feel a beat is to pick a consistent sound in the beat (I.E. Snare, hi-hats, bass drum, etc) and play to that sound, if you're playing with a recording, play to a different element of the same beat a few times, and once you've learned those, you'll be able to switch between the two, even within the same riff. After you get used to 'dissecting' the sound, you'll really be a stand up bassist. Usually in a jamming situation, if you're new to playing with the drummer, they'll want to work something simple out to get a feel for your style (and you, theirs), remember, they're always learning too!
Try to find a drummer (or drummers) in your area to jam with, its always nice to have a live drummer because you can really hammer out a good rhythm much faster than playing solo or playing with a loop, or other recording. Guitarists can be tough to jam with sometimes if you're not used to providing a rhythm for them to build on, generally the drummer locks everything together, but in their absence, you fill that position. To avoid having a lot of troubles with guitarists, learn to watch what notes they're playing and where their hands are heading. Give them a solid line, and once they lock time with you, then you can build off your own bass line and even their guitar riff if you're paying attention to their movements. If they know music theory, have them explain a riff to you step by step, or at least what scale they're using (if you can't tell by ear, another skill to practice!), from which you can build your rhythm if you know the scales, or at the very least you know what scale to look into next time you play.
Mind you all of the above is merely a guideline from what I've seen firsthand, and what I personally do to emulate the sounds of those genres and to hone my playing, to varying degrees of success. The golden rule is to have patience! There will be times were you're going to get frustrated. Don't freak out if you can't nail that crazy solo the first time around, sometimes you have to play it through at a thousandth of the speed a couple times to get it right at full speed. It can be irritating, but slowing that riff down gives you the control you need to get it right at full speed, and you will, if you keep at it!
As far as original writing is concerned... Its called 'original' for a reason!
Try exploring your fretboard. You probably know a few notes that sound good played one after the other, so build from that! For every great bass line out there, there were a thousand mangled lines before it, but each of those lines will teach you which notes sound good together, and which don't. Music theory will do this as well with scales and modes, but theory is nothing without practice.
Good luck, my friend, and good rhythm.
