Bicycle is the standard. Every guesthouse rents bikes for around 50 EGP a day, and the oasis is mostly flat. Cleopatra's Spring, the Temple of the Oracle, the Mountain of the Dead, and Fatnas Island are all reachable on a single day's ride.
Donkey carts (caretta) operate as informal taxis around town. Agree the price first — 30–50 EGP for a short hop is fair.
For desert excursions you'll need a 4WD with a local driver: guesthouses arrange these on demand. Solo travel into the Great Sand Sea is forbidden and dangerous; the sand swallows ordinary vehicles within minutes.