I say hinged doors with the hinge at the bottom so the cage can be easily opened an accessed fully. You could use one of those long single brass hinges for extra support. I also allows for nice secure latches/locks at the top. I would personally use 1/4"-3/8" tempered glass because although plexiglass has better clarity it also gets cloudy and scratches easily.
Sliding I prefer in tighter spaces so you don't have the door in your way. Swinging could give you easier access for spot cleaning etc though.
As for synthetic versus glass, the synthetics can scratch much easier and look like crap over time but while glass doesn't scratch easily and cleans up well it can pose its' own hazards if it breaks.
Ultimately, the choice is yours and you know better what your needs and priorities are.
I prefer sliding doors. Get glass if you can plexi/acrylic doors will scratch up far easier and be prone to warping. If you use plexi/acrylic don't use windex or any other glass cleaner on it, you have to get a special acrylic cleaner.