Egypt is a big country but most of the places travelers visit are along the Nile or the Red Sea coast, and connections between them are decent if not always comfortable.
Trains run the length of the Nile Valley from Alexandria through Cairo to Luxor and Aswan. The overnight sleeper train between Cairo and Luxor (about 10 hours) is a classic way to travel and reasonably comfortable. Book first or second class -- third class is an experience you probably do not want. Egyptian National Railways is the operator; tickets can be bought at the station or, for sleepers, through Watania Sleeping Trains.
Domestic flights connect Cairo with Luxor, Aswan, Hurghada, Sharm el Sheikh, and Alexandria. EgyptAir and Air Cairo operate most routes. Flights are cheap by international standards and save a lot of time.
Buses are the main option for the Red Sea coast and Sinai. Go Bus and East Delta are the main companies. Upper Egypt Travel handles routes to the south. Buses between Cairo and Hurghada or Sharm el Sheikh take 5-7 hours and are air-conditioned. Book ahead during holidays.
Nile cruises between Luxor and Aswan are both transport and attraction. The standard trip takes 3-4 nights, stopping at temples along the way. Hundreds of boats operate this route, ranging from budget to luxury.
Within cities, Uber and Careem work well in Cairo and Alexandria and are far easier than negotiating with taxi drivers. Cairo also has a metro system -- cheap, fast, and useful for getting between key areas. It gets extremely crowded during rush hour. In smaller towns, local taxis, microbuses, and tuk-tuks are the way to get around. Always agree on a price before getting in.
Driving in Egypt is not for the faint-hearted. Cairo traffic is legendary and rules are more suggestion than law. Outside the cities, highways are reasonable but night driving is genuinely dangerous -- unlit vehicles, animals on the road, and aggressive overtaking. If you want to explore the desert or remote areas, hire a driver or join an organized trip.