Just 2 of us traveled as I really don't like big tour groups - just too insulating from the culture. And I don't particularly like throng of tourist crowds and, for that reason, we didn't bother to go up river to Luxor. We also did not get to the White desert, but that is something I absolutely wish we had done in hindsight and I hope to do again. In a nutshell, fly into Cairo, bus to Alexandria, plane to Hurghada (mainland coast of Red Sea), ferry to Sharm-El-Sheikh (Sanai Peninsula), flight back to Cairo. One quick note - unless you have a non-stop flight to Africa, I highly recommend buying separate round trips between the U.S. and Europe and Europe and Africa. Even the exact same flights are a lot less when purchased that way. Plus you get to visit part of Europe. I find Europe to be a very nice, needed recovery point when coming back from Africa. Africa is wonderful, but it can be tough going.
The bus was a pleasant trip and we had fun talking to some locals on that. The ferry doesn't run year round, which was a surprise to me, so I am glad they were running then. It is also the only large ferry I can recall having been on that does not have access to the outside. One Egyptian Red Sea ferry sunk the same day we were on ours.
We stayed at the Ramses Hilton in Cairo, with a balcony overlooking the Nile and the pyramids slightly visible though the haze. There was a belly dancing show up at the top of the hotel around 11pm. Wonderful. The hotel could arrange trips over to the pyramids, but we decided to just go downstairs and hail a cab. (I had done that in Beijing - negotiated a price to take us to the Great Wall for the day and back, and it ended up a fun adventure as we figured out quite a way into it that the cab driver didn't know where the Great Wall was. Time consuming, but fun.) We did the same thing for the pyramids which, although it worked out great in the end for us, I wouldn't recommend as it was a bit uneasy at times. (Another long story that ends great but had its moments, to be sure.)
I saw no evidence that Alexandria gets many tourists. We went there to dive, and that was mind boggling. Amphores from an ancient Greek shipwreck (ship had long since decayed); relics from ancient Egypt, including huge stones that had once made up the Alexandria lighthouse - one of the seven wonders of the ancient world; a submerged floor that was above the sea when the Romans were there; and an intact World War II Italian fighter plane. All of that history on dives in one day. While most of the really good Egyptian relics had been removed, there were still some small sphinx with their heads separated from their bodies. Almost as amazing as all of that is that there was only one dive outfit there and we were the only 2 customers that day. Afterward, we were having fun conversation with some locals. They asked us where we were from. With "America" as the answer, the conversation was DONE. From then on, we were Canadians. Everyone loves them, and we went back to having fun with the locals.
Hurghada was a whole lot of construction towards being a tourist destination. We stayed at a picture perfect resort, but went out looking for local life. Nothing at all within walking distance, except a small gift shop, but we struck up a friendship with the owner. He ended up making us tea and sharing his shisha (hookah) pipe. (English was spoken everywhere in the country.)
The diving in the Red Sea is unreal for shipwrecks and coral. Fish life was decent, but the visibility was spectacular. I recall being 90+ feet down and clearly seeing the surface above me, as well as quite a bit further down. Sharm-El-Sheikh has a wonderful little city center closed to traffic. Palm tree lined streets with nice shops and restaurants. It is unfortunately hit by terrorists attacks more often that other places, which is why I think cars are not allowed in the center. I believe that Russian jet the blew up recently took off from there. Speaking of Russians, there are a lot of them in Egypt as that seems to be one of their preferred vacation destinations. I don't recall meeting any other Americans.
That's a very long nutshell of the trip. Again, I definitely want to go to the White desert. I heard wonderful things about it from people we met in Egypt. We're thinking of a trip to Turkey soon with Dubai and possibly Jordan mixed in. Perhaps we'll add that to the itinerary.
Best wishes with your health