During my free time in Granada, I did many different things.
I was lucky enough to stay for almost a year (11 months), so I was able to discover the region (and the country) when the weather was good (or even very good !!) but also when it was cold in winter. Yes, Granada is located very close to the Sierra Nevada mountains, at about 800m above sea level...
So I did a lot of hiking (accessible by bus or we sometimes rented a car with friends): there are great places to run and/or walk with beautiful landscapes, in summer and winter.
I was also lucky enough to arrive the year the ESN section was organising a lot of open mic in a night club. We were a whole group (the "erasmusicians") of international and Spanish people playing music ! It was really nice ! We rehearsed in parks, in a music studio, but also outside the city, at the Mirador San Miguel Alto (for me, the best viewpoint in the city where I spent a lot of time admiring, relaxing...), by the sea, etc.
I also went to the sports sessions that the university offers (tennis, football, dance, and many others), I recommend ! Because you meet local students who are not necessarily close to the international world and so you meet different people and you can learn more about the local culture :) Because my goal was really to integrate myself as much as possible in the local life !
In Granada, there is a tram/metro and several city bus lines that allow you to get around the city. But the best way (and the way I used the most during my semester there) is to walk! You can look up to admire the architecture, stroll through the small pedestrian streets or climb the stairs of the Albaicin to reach the mirador of San Nicolas and enjoy a breathtaking view of the whole city. What I liked best was walking on "el paseo de los tristes" where the view on the Alhambra is incredible!