Yes, def. stay in Cairo, it is much easier to get around to the various sites this way, as is the case for most other large, sprawling cities. Cairo is a massive city (something like 18 million population) with many neighborhoods, and a vast suburbs as well - you don't want to spend too much time navigating to get around the city, depending on what you plan to see. Last time I was there, we stayed in a furnished apartment in Zamalek. Zamalek is on the island of Gezirah in the centre of the Nile in Cairo - very nice area and a lot of foreigners stay there. We struck a deal with a cab driver and had him drive us spots every day - pyramids, museum, bazaar, etc. which wasn't too expensive and worked out very well. Or you can do bus tours. Don't ever think of renting a car to drive your self around either. People drive like maniacs!! I recommend a furnished apartment if you're like me and get sick of eating out everyday. They are not too expensive at all (I've seen many advertised for $50 and up per day and in very central areas in Cairo.
Regarding Hurgada - it is many miles away - on the Red Sea in fact, and no where near Cairo. It's a resort area famous for its amazing scuba diving. Hope this info helps.