You can use whatever you want as a lining- I've used pigskin, lambskin, garment-weight cowhide, and elk. Lambskin isn't the strongest leather out there so if you use it, make sure it's especially well adhered to the strap all over, but it is nice and soft. The guy I did the elk one for still uses his so I guess it's held up all right!
If you play guitar yourself, you'll be able to use your experience with straps for reference. I don't but according to my guitar-playing friends, the thing that makes guitar straps comfortable or otherwise is the suppleness of the leather. So when you're picking out your tooling leather, try to find something you wouldn't mind having slung over your shoulder for a few hours. You don't have to use really heavy leather; especially if you plan to use a lining, you can probably get away with lighter leather than you think. Also oiling it can make a big difference toward suppleness, and if you want an English-bridle look, just give the strap a few coats of oil (peanut oil is cheap and gives a nice color) after you're totally done tooling it and lay it out in the sun for a few hours. The more oil you use and the longer you let it "bake," the darker and more flexible it'll get, just don't completely drown it in oil- over-oiled leather is not nice. Finally, for wider straps you might consider adding some foam padding (you can get 1/4 inch stuff from Tandy) where it goes over the shoulder. You have to fiddle with it a little to get it tapered off right so the lining doesn't have weird ridges in it, but it feels really nice.