As Granada is a compact city walking is the easiest option, however, there is a good bus system if those comfortable walking shoes are not as comfortable as you thought, or if you want to get to many of the attractions in one day.
Buses run to the Alhambra from the commercial centre throughout the day and a single ride will cost approx. 1€ but you can get a booklet of 9 tickets for 5.50€ or one of 20 tickets for 11€. You don’t have to search for a ticket office as you can buy these as you board the bus directly from the driver, but make sure you have the right currency and have smaller denominations if you don’t want to cause a bottleneck. Your best choice is bus 32 if you want to get to the Old quarter or bus 30 which runs from Plaza Nueva. Bus information
Taxi is a more expensive way of getting around, but if you prefer it, most city short hops will cost you in the region of 8€. You can hail taxis in the streets just as you do back home. You will easily find taxis outside the major hotels and in the Plaza Nueva.
Not for the faint-hearted as Granada suffers badly from traffic congestion, also navigating the narrow streets of the old quarter can be hair-raising, to say the least, and then there is the problem of parking. If you only need the car to arrive in Granada or to leave Granada you will manage, but to get around within the city it really is not advisable.
In some neighbourhoods, it really is the only way to get around and most especially in the Albayzin and the Alhambra.