What's the best way to travel from Granada to Madrid. Ideally we'd like to take the AVE (what connections would be needed?) but would also consider driving.
Actually the best could be a bus in Spain. Buses in Spain are great - cheap and reasonably quick. Just check the schedules and see what fits your schedule best.
Trains are usually direct from Granada to Madrid P. Atocha station, but there is currently a replacement bus on the portion from Granada to Antequera-Santa Ana, due to construction on the train line.
There is no AVE - which is the high-speed rail service. The best train is a slow train with track construction requiring a bus and transfer. The ALSA bus is an option. Or fly. You should be able to get a pretty cheap flight.
The bus is about 5 hours - same as the train. By the time you factor in the additional time to get to/from airports and airport procedures (and probably having to check luggage), you don't save much - if any - time by flying. The train is much more comfortable and less hassle. The bus is comfortable.
Drive/take a bus from Granada to Cordoba (bus company: Alsa), then take the AVE in Cordoba to Madrid (train company: Renfe).