IIRC, I cut the aubergine into ~1.5 cm thick, round slices. I brushed the two flat surfaces with a bit of oil and then fried them. Then, when they had cooled a bit, I chopped them up. I don't remember for sure, but I think I left the skin on.
For the baba ghanoush, I roast the whole eggplant in the oven after having poked holes all over the skin so it won't 'splode. It should collapse and might even have some very dark or burnt bits on the skin. Let it cool and then scoop out the very soft flesh. No skin in baba ghanoush. The recipes I have seen have you put it in a food processor, but I don't see the point, as long as your garlic is finely grated. Just stir in the tahini, garlic, lemon juice and any herbs or spices you want to add, like a bit of cumin. I have read you should drizzle it with some EVOO and that a shake or two of sumac is good. I have yet to try it with sumac.