lets see all the options.
bus: you have to follow their timetable etc. for a bus from agios-elounda again you have to make sure there is a connection and same for the way back
taxi: costs a fortune and its pointless as when in agios you will have to find another way of moving to elounda
bikes+quades: DO NOT, i repeat DO NOT use quads of the main road. i have seen many accidents/deaths due to those death traps. better take a bike.
car: by far the best option for me
now trip to elounda. as someone else said there is a boat going from elounda to spinalonga very often. from agios most of the boats depart at around 12 and are back 4-5 depeding weather conditions.
roads in crete. North crete is well developed even though we still have some problems. from where you will be all the way to elounda you will have no problems at all.