When someone tells me they're traveling to Spain and looking for recommendations, San Sebastian (Donostia, in Basque) is my first thought for city destinations in the northern part of the country. It's charming, beautiful, and steeped in history. Mount Urgull, the hill that stands on the right side of the bay, epitomizes all those characteristics in one easy to explore package.
Mount Urgull is an easy walk from anywhere along the Promenade, and you can either walk up from Parte Vieja or walk around by the aquarium and the point to see views of the Bay of Biscay. I'll recommend that you start with the second way and, coming back down, drop right into Parte Vieja. That way you get the best of both.