The Jerusalem Old City is a major area of importance for Jews, Muslims, Christians and Armenians and has 2,000+ years of history jam-packed in one format.
The 0.35 sq miles are so overwhelming that almost every building is a museum or of some historic importance. Besides all the religious importance, it also used to be a fully-built Roman settlement.
For my second visit to the Jerusalem Old City, I opted for another abbreviated experience to take in the sights without an organized tour.
I went up to the Ramparts Walk and made my way through the city towards the Wailing Wall, criss-crossing many of the old alleys.
Note that almost everywhere in the Old City (and in most of Jerusalem) I felt extremely safe. At least right now it is easily on the level of, say, San Francisco. There are tour groups but the Old City never felt crowded. Security and military presence was negligible at this time. It IS an excellent sight that you should not miss.