I lived in Cairo, Egypt for a period of time in the early 90's. While I'm sure much has changed since then I'd like to recommend the following for your must see/do list:
For an amazing shopping experience the immense rabbit-warren of shops and stalls in the mostly outdoor shopping district the Khan al-Khalili (a.k.a. the Khan) can't be missed. Originally built around 1382. Take a map of the Khan with you...it's easy to get lost in there. You can also hire a guide but he will most likely only take you to the stores of his relatives, still if you don't have a lot of time to spend it's an option to consider. Bargaining is an artform in Egypt so practice your skills. If it's something you REALLY want try to act uninterested and don't be afraid to walk away. You can get some excellent perfumes custom made (they will copy designer scents as well). Frequently the perfumes come in pure oil form and will last forever. You can also get some excellent buys on gold bangles and bobbles! For big ticket items be prepared to spend a considerable amount of time negotiating (often involving sitting down to tea) before closing the sale.
While you're in the vicinity you might want to take a taxi or bus over to the City of the Dead, the City of the Dead houses the funerary complexes of the Mamluk sultans and their amirs. There are some amazing Khanqah's there. The khanqahs supported large numbers of inhabitants who developed complex communities. Some of the more notable: Kanqah of Barquq, Khanqah of Sultan Barsbey and the Funerary Comples of Qaytbey.
Old Cairo is also worth exploring. The easiest way to get there is to take the Metro south (Helwan direction), get off at the Mari Girgis stop. Plan at least a 1/2 days worth of exploring here to truly appreciate all the amazing architecture and history (the Coptic Museum is there).
The Egyptian Antiquities Museum is a must while in Cairo. Not something you can do in just one day. This is a vast collection of Egyptian antiquities.
Of course you have to stop at the pyramids while you are there. We enjoyed renting horses rather than camels whenever we went to Giza, the horses seem to attract less tourists, and you can find a personal guide for a small party of 2 or more.
If you want to stay out of the central hub-bub of the city center, Maadi is a good option for accomodations. Maadi is located in the southern suburbs of Cairo, there are some gorgeous villas there left over from the English settlements. The hotels can sometimes be more affordable in Maadi and New Maadi. Taxis (find one driver you like and stick with him -- negotiate day rates) and the Metro provide easy access to all other areas you might wish to visit.
Other places to visit in Egypt: Alexandria, Memphis & Sakara, Luxor, Aswan and Abu Simbel. For a real adventure consider renting a Falouka and sailing up the Nile, not recommended in the hottest months, bring lots of mosquito protection.
The Siwa Oasis is a must see destination. It is one of Egypt's most isolated settlements. While it has become more of a tourist draw in recent years, it's still far enough off the beaten path that you're sure to have a wonderful adventure and have a chance to get a closer look at this unique Siwan Berber culture. Don't miss the fine crafts produced by the locals. While you're there consider a Siwa Safari.
There's so much more to do and see, but those are some of the highlights.