There are two old parts of Cairo, the Coptic Christian neighborhood and the Islamic neighborhood (known as Old Islamic Cairo). The former is a pedestrian-only area which you can access via the Mar Girgis metro station. There's a lot of security on Saleh Salem Street and there's usually a police checkpoint with a metal detector; if you go by the metro, you won't have to deal with it. The points of interest include the third century Hanging Church with its beautiful icons from the eighth to eighteenth centuries, the ruins of the Roman Bablyon Fortress, the Greek Orthodox church of St. George, and several lesser churches. The cemetery is also nice. It is a tourist area where there are not many local places to eat, and the prices are higher than average.
The most interesting thing about Cairo, not including its pyramids, is something you'll find in a place called "Old Cairo." There is where the real essence of the city is. To pass through its streets, losing yourself among the markets and even learning to haggle, is one of the experiences that's not to be missed. It's also interesting to be able to go in a mosque, even though, of course, women have to be covered. Normally they don't let tourists in, but there are some the are ready for foreigners that they can go in if it's not during prayer hour.