The airport is about 25km outside of Granada.

There are frequent airport buses.
The price of the airport bus is about 3 euros per person.

Most of the buses go down the Gran Via, then go down towards Puerta Real and then past the Corte Inglés. If you want to go to Plaza Nueva you would have to get off near the Cathedral.

If you don't want to be walking in the street with your luggage you can catch a taxi.

The official price of a taxi from the airport to the centre of Granada is 25 euros.
However taxis rarely charge the official price and the price is usually between 25 and 28 euros.

I have lived in Granada for many years and taxi drivers do not generally rip people off.
However, recently it is becoming more common for taxis from the airport to charge extorsionate rates from the airport such as 35 or 40 euros.

I have contacted politicians, the police and the body which controls taxi fares in Granada about this.

The best advice if you really don't want to be overcharged is this:

If you don't speak Spanish write this on top of a piece of paper:

¿ Cuanto cuesta ir al?
Write your destination address here.

Hand it to the taxi driver with a pencil.

He will then write the price for you.

If the price is reasonable ie less than 28 euros, get in.
If not then you have a perfect right to reject this taxi and go to the next one in the queue.

The organisation which controls the taxis says that if you wish you can insist on being quoted the fare rather than using the taxi meter on journeys from the airport.