By Plane -- There are strong rumors that the nearby Israeli-built airport may open soon, but there have always been rumors to this effect.
By Car -- If you're coming up from Sharm el Sheikh, a rented car makes good sense. It may be more expensive than the bus, but the freedom of moving when you want and having transport around a taxi-less town is worth it. Moreover, thanks to a European Union aid package, the roads are exceptionally good here.
By Bus -- There is a new and fairly pleasant bus station at the western end of town. The East Delta bus leaves for Cairo at around 6pm. There is an earlier bus that goes to Suez at 1pm, and a bus to Dahab and Nuweiba at the same time. Tickets cost LE40 ($7.30/£3.70) Cairo, LE30 ($5.45/£2.80) Suez, and LE16 ($2.90/£1.50) Dahab.
If you've arrived by bus or taxi, you'll quickly find yourself stranded if you haven't made arrangements to be picked up. With no public transport or taxis, getting around St. Catherine is a matter of prearranging transfers or hitching rides. The latter is surprisingly easy -- there are usually foreigners staying in town who are heading up to the monastery in their own cars.
There is a Banque Misr beside the gas station in town that's open 9am to 2pm and 5 to 7pm; it's a good place to change money. The Internet cafe Desert Net has six clunky PCs available for LE6 ($1.10/55p) an hour.