The answers will be different for everyone. For me, it comes down to four factors. Top priority is a good transmission, whether manual or automatic. After that are communicative steering, sufficient power, and a nice balance between handling and ride comfort.
I've rented three different cars in the last 18 months. Previous-gen Civic, current-gen Sentra, and current-gen Malibu. I hated the Civic because the steering wheel had all the feel of a living room dimmer switch, and the transmission acted like it was filled with marshmallow topping. It had sufficient power for accelerating, though because of the steering I never got comfortable with the handling.
The Sentra was far superior. The CVT adapted to my driving style, power was adequate, I could tell through the steering wheel what the front tires were doing, and 8/10ths handling was a blast. On some very curvy roads in the Santa Cruz Mountains it took everything I wanted to throw at it, to the point of nearly overheating the brakes. I wouldn't mind owning one.
The Malibu fell short with its transmission. There was an annoying delay during upshifts, no matter how hard I was accelerating. The steering, handling, and power were fine, but the tranny was a deal breaker.
My 2000 Protege has become less fun since I swapped out the transmission (failed diff bearing, I think) for one from a junkyard. The syncros in the junkyard tranny are are nearly worn out, so shifting is slower than it should be. I still have the Eibachs and Tokicos on it though, so not all is lost. I'll rebuild the tranny here soon.
I haven't had the P71 long enough to decide if it's fun or not. This is the first big American V8 car I've owned since I had to let my 65 Impala SS go way back in 1992, before I knew much of anything about cars.
I'll leave the Miata off this list since it's so obvious.
All this being said, after 22 years I still miss my old 1980 Toyota 4x4. More than the Impala, in fact. It had good steering, but the worn-out engine made maybe 60hp, the manual shifter was as slow as an M. Night Ohgodnotagain movie, and it rode like an empty Kodiak box truck. I wish I still had it.
Dang, this turned into a tome. Sorry.