The Parte Vieja (Old Part), usually called "Lo Viejo" by the donostiarra owns two churches, the Constitution Square and more bars you could ever imagine. The streets are pretty narrow and cars are only allowed in exceptional cases, most of the buildings are from the 19th century. This zone is pretty touristic as you might expect.
This street called Juan de Bilbao is the only one of Lo Viejo that isn't touristic at all.. a friend of mine called it jokingly "Calle de la ETA" (ETA street) but don't say it out loud there! You might rather stick with the expression "abertzale" (which is a synonym of the Basque nationalism) or the real name. If there is any pro-ETA demo in Donosti it's here and they prefer talking in euskera, the Basque language, too. Also you should consider taking pictures twice for it's not really approved. Still, don't hesitate to explore this street as well!
It gets pretty crowed as soon as the sun starts to show