The above posts cover most of the important sights. There's also a restored synagogue in Cairo that's well worth a visit, regardless of your religion. Same with the big mosque whose name I forget (Ali something?).

A more primary decision, though, is one city or two (I can't imagine trying to do three in five days). That will depend on what kind of traveler you are. If you're content to do the highlights only (i.e., pyramids and Egyption Museum in Cairo, but precious little else), Then do Cairo and Luxor (including the Valleys of the Kings & Queens). Adding Karnak may be pushing it, but then again it's close to Luxor if you find you have the time.

But if you're the sort that would rather see a location in depth (e.g., the mosque, the Coptics, the synagogue, etc.), I suspect that Cairo will be plenty to fill your time...and leave you wanting to come back for Luxor / Karnak / Aswan / Abu Simbel.

Have a great time. It's definitely third world (with all that implies, for good and bad), but also definitely worth it.