Granada is a small city; everything can comfortably be reached by foot.
By bus
Buses old stylish US or Canadian schoolbuses go just about everywhere at about every time, you see them and if you slightly look like anybody wanting to go anywhere, be sure they'll load you on their bus. Another option are the mini busses which have a bit more set time and you can book in advance, they're more comfortable but cost a bit more.
By ship
Granada's islets are not to be missed, and the way to see them is by boat. Boat tours leave from Puerto Asese, about 5-10 minutes from downtown by taxi.
By Coche
Horse-drawn carriages, known as coches, are something else, ancient and useful and found everywhere but often the horses are so worn out that you should seriously ask yourself if you want to contribute to their suffering.