I've used the Mickey Baker book for learning some basics in jazz on guitar. It is an extremely difficult resource even on guitar (I can't imagine how difficult transposing it would really be).
A good thread and musical development site for jazz is
www.jazzguitar.be. I've been there for a while and I learn a ton (and there are a few ukers hiding in the forums).
I could boil it down to you needing to learn two things to start your way into jazz:
1) Comping - Better known as accompanying. This involves learning how chord voicings interact and how rhythm creates a background for a soloist.
2) Soloing - Not exactly limited to improve.
For the first, I would suggest getting some chords to Satin Doll (another good one is I've Got Rhythm) and playing around with it until you can comfortably play the song from beginning to end and sing along. Make sure you try really hard to play the extended harmonies (you know, the chords that have things like "9" or "13" on them). There are plenty of resources out there that can help you learn how to create those chords.
For the second, you need to first learn to play a melody. Try Satin Doll again and just play the melody. Then, start adding ideas (or "motives") to the line and try to get farther away from the original melody before coming back. It will take a lot of practice to really develop your ear for this, but it is totally worth it.
Also, you may want to record yourself or otherwise find a way to have someone comp for you. I like using electronic resources for this since I'm shy when I learn. The most popular one out there is Band In A Box (BIAB), but it is expensive and a little overwhelming to start with. I use a couple of other ones that you can find around. One of the best free ones is
www.djangolizer.ch. It is web based and mostly free.
In the long run, learn your music theory. It will help. You don't necessarily need to learn to sight read music notation, but it can help. Keep at it and have fun.
~DB