Most of Hurghada's best sights for your holiday are out to sea. Enjoy a boat trip out to the Giftun Islands where some good diving and snorkelling is available. If you're feeling adventurous, there are several wrecks to explore, such as El Mina and Kimon M, which are even shallow enough for novices.
Otherwise, you could climb the Anfish Mountain just on the outskirts of town, it's not particularly high so it won't prove too strenuous but the views across the town and bay are impressive. You could also explore the old fishing town area around Dahar. Its main mosque is Saudi in style and quite striking.
Whilst on holiday in Hurghada, take a trip into the desert interior where you can explore the ruins of ancient civilisations among the arid landscapes. Sites include Roman quarries Mons Porphyrites and Mons Claudianus, and the 3rd and 4th century monasteries of St Paul and St Anthony.