At some point next year Granada might be connected to a high-speed railway. But until that happens, at present I think it's a toss-up choosing between coach ("bus"), and train between Madrid and Granada. The former is direct, fairly comfortable and, usually, faster/cheaper. The latter is a bit more comfortable, but always (I think) involves a change if using faster services. You can walk about on the train, which you can't really do on the coach. Granada's coach station is a little further from the city centre than the railway station - though both are far enough out to justify a taxi (in my view anyway). If you choose the coach, then I think the operator is Alsa and there may be a "premium" service as well as the standard one.
I would only hire a car if you were planning to make a "road trip" of it - at least spending the day en route and stopping along the way, perhaps overnight. You do drive near some great stops if that were of interest. But if, as I presume, your interest is just moving from city centre to city centre then I don't see any advantage in driving (perhaps price?).
Pretty much the same comments for the Granada-Sevilla leg. As its a fairly short journey, I'd use the coach to avoid the current need to swap transport on this route if you go by railway (again, until the new line opens).
For Sevilla back to the capital, I'd definitely use the high-speed AVE train, at least for the stretch after visiting Cordoba.