It depends of your arrival city. If you will arrive to Hurghada you can take bus from Hurghada to Luxor , then train from Luxor to Cairo , then bus from Cairo to Hurghada.
If you will arrive to Cairo you can do this rout vise versa.
Generally from Cairo or Luxor to Hurghada use bus.
Cairo to Luxor use train.
If you have high budget you can use flight.