Cairo has many sights, enough for a week or two. Here they are divided into sections according to their nature.
Of all the sights you cannot afford to miss the Pyramids (would you consider skipping them?), the Spinx and the Egyptian Museum. They are the unique remainders of the oldest civilization of the planet.
Try to get a good taste of islamic culture as well by visiting some of the mosques and the old town. Visiting these sights will give you some understanding of the basis of present day life in Egypt. Furthermore the COptic quarter offers some interesting sights worth exploring. The city of death is a category apart and merits a visit as well.