Philly and LA, also--not easy instruments to learn, because you have to learn completely different scales, pitch, etc--Also, Arabic music and Turkish music are totally different, so you have to choose--contemportary belly dancers tend to prefer Arabic music--though a while back, it was mostly Turkish that was preferred--

An easier idea is to learn a few Armenian tunes--they use the Western scale, and they even use electric guitar from time to time--

For a start--I suggest you learn to play Miserlou, it is a standard with every Greek/Middle Eastern Club band, and familar to all Belly Dancers--

Oud is a very cool instrument, but has a very high threshold of competence, because you have to develop an ear for the music, a feel for the beat, and master the instrument, and you have to learn to improvise--

My best advice is to buy a dumbek, and get someone to show you a basic rhythm pattern, you'll be able to play for your dancers right away--also, you'll need to know the rhythms while you are learning to play the oud--

I warn against the baglama--the care and feeding of this instrument is a real pain, it needs amplification, and god help you when you break a string(or even need to re-tune)--

I assume that you are a guitar player--I have worked out a guitar tuning that allows you to play Eastern European music, Greek, Armenian, etc, with a very oud-like feel--it is similar to Turkish Oud tuning (but with six strings instead of 5 courses)--it requires a bit of explanation to get it to work, but it does work very well--let me know if you are interested, and I'll put pen to paper (I haven't written anything about it, so it'll take me a bit of time--otherwise I'd have just posted it here)

Good luck--