By ship
Ferries run regularly from Nuweiba across to Aqaba in Jordan, bypassing Israel and the sometimes complicated border arrangements. Generally there is no visa fee for entering Jordan through Aqaba since it is a part of the free trade zone. The line to Aqaba is operated by ABMaritime, see their website for the 'official' timetable and current prices.
There is more information about the boat crossing in the itinerary Ferries in The Red Sea
By bus
Buses are operated by East Delta coaches. This is basically a shabby bus that goes to many places including Cairo, Dahab and Sharm el Sheikh. The prices are very cheap about 50p but the buses are really not that good. They are ok if you are going to sleep a lot of the way. It is not just a bus for tourists it is also a local bus service so ladies be sure to cover up to a certain amount.
The bus trip from Cairo to Nuweiba takes at least 7 hours. Don't believe the information about 5 hours - it's 670 km, two stops for toilets and tea, several stops for taking up and dropping passengers and at least two military checkpoints at the coastline where they check the passports and visas. So don't forget to bring it with you, identity cards will bring you into a short police interview the bus will wait for you fortunately.
Tickets form Cairo should be booked in advance, the central bus terminal is called Turgoman and is located near the Ramsis Railwaystation. Unfortunately you can only book one-way. Tickets from Nuweiba are sold at the East Delta Company bus terminal. No reservation is possible. If you stay at a camp between Nuweiba and Taba, you can simply wait by the roadside and wink the driver to stop. You will then be able to purchase your ticket during the short stop in Taba.
By bus to St. Catherine: as of the end of October 2011, there is a regular, twice weekly, minibus service to and from St. Catherine. The Bedouin Bus runs on Wednesday and Sunday and costs 50LE each way. Check the website for details on pick-up and drop-off points in Nuweiba and St. Catherine.