Visit cairo for pyramids, Egyptian museum, Egyptian tower and castle.. I think 1/2 days for cairo is enough because it's kinda busy and crowded!
You can also visit Alexandria it takes 3 hours by car/bus/train from cairo, for 1 day as well for the Bibliotheca Alexandria, Qaitbay citadel, Roman museums and Montazah parks.
Luxor and aswan is a must they are far away from cairo and alexandria for the great ancient museums and temples also the great nile scenes, it would be kind hotter than cairo and alexandria because it's in the south, both of them are next to each other.
If you are seeking the great beaches, corals, underwater treasures, head to the red sea, "Sinai" to be more specific! you can visit, sharm el shikh, Dahab, taba, marsa alam, sahl hashesh and hurghada, you will enjoy diving, and a lot of sea activities there!
These are the most known cities in Egypt for tourism and you can google names for more details and info. Welcome anytime
