For the outer, some of the leather types that are popular & easily accessed at the moment are:
* Shell cordovan
* English bridle from Sedgwick/Clayton/Baker
* Natural vegetable tanned leather
* CXL from Horween
My preference would be for a high quality natural veg tan. None of the other leathers will age with as much grace or character, IMO, compared with a simple but good quality veg tan. Anything from Wickett & Craig carving leather to Buttero or Minerva Liscio would be a nice introduction to leather!
Shell is popular, but I think it is somewhat of a 'superficial' material - not nearly as magical as grained leathers; best used on footwear.
CXL can work well, though it dirties easily, and takes a while in terms of patina development before it looks good.
English bridle lends class, though in my own experience they don't age with the some grace as a less processed or raw veg tan...IMO, more for people who want things to look new, rather than for folks who want a bit of aesthetically pleasing wear & tear.
For more 'elegance', you could try a finer leather, though I don't think they will suit the bifold you have in mind:
* Chevre goat
* Baranil/Barenia calf