The simplest way of getting around Egypt from one major city to another is by air, and EgyptAir have good connections on most main routes. To see the sights around a particular place, the best way of moving is by taxi, although the pyramids around Cairo, and the tombs at Luxor are well served by guided tours by coach.
You can hire a car in Egypt and self-drive, although the country is quite large, and the distances between cities are long and fuel stations are few and far between in the desert.
A relaxed way of seeing the country is to cruise up the Nile; there are many different boats that offer this, from the basic to the luxurious.