This quarter has a great atmosphere and "Spanish" feel to it, lots of winding paths and interesting buildings. Walking down a high-walled cobblestone path, hearing Spanish schoolchildren playing somewhere out of sight, or standing in a tranquil mosque overlooking the Alhambra, were very memorable experiences! There's several lookout points too.
If you're determined you can go straight to the top of the hill, among the dodgy gypsy caves, and treat yourself to a truly stunning vista of Granada and the surrounding area. Bring good footwear as it's all uphill/downhill and the very bumpy and uneven.