While in Cairo spend a day in Giza at the Sphinx and three main pyramids. In Cairo itself, the Cairo Museum is a must see and the Citadel is pretty interesting. The Coptic quarter is cool to walk around (just don't wander out the back into 'the suburbs' - there is a not very visible military facility nearby and the police don't like strangers walking towards it...). Many other areas of the city are worth exploring as well, but be prepared, it's very big and the traffic is insane. If you take a taxi, be sure to know the going rate, they don't use meters so you bargain before hand. You can ask the locals and they'll tell you what they pay, but expect to pay a little more.
Luxor, of course, is the biggest tourist attraction. You can fly there from Cairo. The sites are amazing but the town itself is a bit of a drag - you are pestered nonstop by people trying to sell you something.
I'd also suggest taking a short safari and spending at least one night in the desert. I've NEVER seen a night sky like that and it was very amusing to have a desert fox circling our campsite as we ate dinner.
You might like Alexandria as well. It's full of decrepit colonial architecture, has some good sights including some catacombs. It too is great to wander around. It takes the better part of a day each way from Cairo.