By bus is the easiest way. They are comfortable although the journey is a little under five hours. It will cost you around 16 Euros to La Linea, which is the border town and you can book online here
http://www.continental-auto.es/continental/ventbi.php The border is just five minutes walk from the bus station.
If you don't mind driving, then it will take you less than three hours and give you more freedom. The roads are good from Granada, with dual carriageway all the way. You won't have a problem, although it sounds like you're more dangerous than the Spaniards, whizzing along at 140!!
I would certainly leave the car outside Gibraltar and walk across, as the queuing time to get across in a car at the moment can be over half the time it took you to get from Granada!!