Why Travel to Egypt?

***Cairo and nearby:
The Sphinx and Pyramids. These are sensational sights and shouldn't be hurried. Take a couple of days if possible, climb inside a pyramid, and have a camel ride, horse ride, or walk into the desert on the far side of the pyramids.
There are no café facilities nearby so bring plenty of water.
The excellent Egyptian Museum is worthwhile even for those who don't usually venture inside a place like that, to see the incredible pharaonic treasures, such as King Tut's gold chair or ceremonial hat, well-preserved statues and much more.
 

Khan-el-Khalili is a fun, labyrinthine market with a stupendous selection of excellent souvenirs. It's definitely a must haggle situation!
More Pyramids? There are plenty more smaller, earlier pyramids a little south of Cairo at Saquara, including the famous Step Pyramid. These are easily accessible by taxi.

Cairo needs at least three days, and it's a longish way north of Luxor.
If time is no problem a trip out into the western desert to Siwa or Faiyum oases will be fascinating, though a bit out of the way.
 

***Alexandria, a couple of hours north of Cairo, was once a world famous Mediterranean port and resort but is now a tired shell of its former self and has no good beaches. Good sea food though.
Head towards Libya from Alex and sensational beaches soon appear, especially around Mersa Matrouh, though facilities may be extremely limited, so bring everything you need.

***Luxor is a world-beating spot for a variety of magnificent temples and tombs, including the Valley of the Kings, Colossi of Memnon, Hatsepshut and Luxor Temples.
Stay on the west bank if possible [hotels are small and familiar there] and/or travel around the temples before/after the package tourist rush to try to absorb the calm power of the places out of tourist focus frenzy.

 

Taking a sailing boat [felucca] onto the Nile [with Egyptian skipper] is a pleasant way to escape the crowds, especially for sunset.
n.b. The town riverbank is being ruined by  cruise boat overload. If you care for the environment, or for your health, avoid these noisy, poisonous monsters unless they have alternative power sources.
Luxor needs at least three days and is a few hundred miles south of Cairo, so will need a flight or overnight train journey [recommended].

***Aswan is a beautiful place for boating on the Nile and laid backism, though it hardly deserves time if you're pressed.
There are no monuments here but it's on the way to the Temple of Ramases II at **Abu Simbel [a 180 mile road trip or short flight].
Aswan suffers a similar problem to Luxor regarding Nile tourist boat overload, though a boat trip down the Nile from Luxor to Aswan, stopping off at a couple of wonderful, monumental temples on the way is a good way to experience the Nile.

***The Red Sea:
*Sharm el Sheikh [Sinai] is a classic, posh beach resort without much ethnic style but comfortable and with superb snorkelling and diving near the beach.
Neo-hippy

*Dahab, 85km [53mls] north is terrific for low-cost style, scuba, snorkelling and for chilling out man, but doesn't offer much in the way of beaches.
 

*Hurghada, on the other side of the Red Sea is brash, modern, fast food and package tour hell, distant from public beaches, with no connection to Egypt at all, though comfortable and sunny - naturally. i.e. not a place for independent travellers.