What is the best way to travel between these two cities? I do not find train service and will not be renting a car. Many Thanks.
Train to Murcia (Renfe Cercania) then bus to Granada (ALSA)
Total time 5.5 - 6 hours.
There are trains the whole way, 11-12 hours with inconvenient connections, but not attractive ones
Gail,
You also have the option of taking a bus from Alicante to Murcia then another bus to Granada.
This is the official website for the ALSA Bus: http://www.alsa.es/en/