The Sinai is a peninsula, belonging to Egypt, but geographical already to Asia. The South of Sinai with the Red Sea is quite popular for tourist, especially Sharm el Shekh which is mostly crowded of Russian tourist and very expensive. Better places to travel for individual backpackers are Dahab, Nuweiba, Tarabin and St. Catherine which is located in the Middle of the mountains of Sinai, next to Mt. Sinai where the prophet Musa got the Ten Commandments. The North of Sinai is not recommended for individual travelling. If you try to hitchhike there the police will probably stop you at a junction and send you back or another way. If you really want to visit North Sinai go with local people or take one of this cheap service taxis.
Hitchhiking is possible, but not easy. There are a lot of taxi drivers all around which try to offer you there services. However you'll be always able to find some friendly Bedouin guys, police officers or internationals who do take you. Always make sure by entering a car that you don't want to pay. 90% of cars/minibuses will stop and around 40-50% will agree to take you for free. "Leysa filoos" means "no money" in (broken) Egyptian Arabic.
A hitchhiker's biggest problem on the Sinai won't be hitching a free ride but rather the police, especially near touristy areas. Apparently, the police are tasked with ensuring all tourist operators and taxis are licensed and pay taxes for the privilege of taking foreigners (Egyptians hitch very commonly for little amounts of money), so they are not very happy to see a foreigner in regular Egyptians' cars. The best strategy to avoid trouble for you and your driver is to ask your driver to tell the police that you have no money and that he's only taking you to the nearest bus station. You could also ask your driver to drop you off just before a checkpoint and then just walk through and see what happens.
It is quite easy to navigate because there are not many roads. There is just this one road going South-West from Taba over Nuweiba, St. Catherine junction, Dahab junction to Sharm el Sheikh and from there North direction Cairo. On the main roads there are also bus connections. Between Taba and Sharm el Sheikh over Nuweiba and Dahab the bus goes twice a day. So if you don't get a ride at all you'll be able to take these buses (e.g. 30 LE Nuweiba-Dahab). There's a new Egyptian bus company, which has a pretty decent website which accepts foreign cards, called "GoBus" - they have 4-5 daily buses from Dahab to Cairo at around LE120-300 (depending on how luxurious the bus is), so you can use them as plan B or if the police prohibit you from hitchhiking.
To get to St. Catherine you go North from Dahab in direction of Nuweiba until the junction to St. Catherine. Wait at the police station for a car which is going all the way to St. Catherine that you don't stuck in the middle of the desert.
Things to do/ to see
The mainsight in South Sinai are the desert and the sea. You will find a lot of amazing spots for diving and snorceling. To go on a desert safari trip with a bedouin guide is an amazing experience but quite expensive. If you want to explore the desert on your own look for a map for the area around St. Catherine and Jebel Musa (Mount Sinai). There are a lot of awesome wadis and mountains and a lot of hospitible and helpfull bedouins, also water is not such a big problem in winter/spring/begin of summer because there are a lot of springs and hidden wells. However it is allways better to go with somebody - it is worth!
How to get to South Sinai
By car:: There are busses from Cairo to Sharm-el-Shekh and St. Catherine (around 50-70 LE one trip), also special offers like Dahab-Luxor etc., anyway best is over-night because over the day it is too hot and most of the buses don't have airconditioning. I have no experiences in hitchhiking this way. Most of it is lost desert. It doesn't seem like a good place. But maybe you are lucky and you find a truck-driver near Sharm who is going all the way.
From/To Israel: The Taba Border Crossing is today the only open border crossing between Israel and Egypt. It is easy to find a ride to the Taba junction (5 km away from the Taba Border Crossing), the rest you have to walk or take a taxi. If you arrive in Taba from Israel don't get confused from the taxi drivers who offer you their services or even lie and say there is no bus any more. There is always a bus at 15:00 going Taba-Nuweiba-Dahab-Sharm! If you are leaving to Israel you will find on the other site a city bus to Eilat. From the central bus station it is close to the main road heading up North to the Dead Sea and Jerusalem. You will have to pay an exit fee of 100 shekels to the Israelis, unless you hold a valid, printed booking to Movenpick, Hilton or Radisson Blu hotels in Taba - you can get a free booking through Booking.com, but keep in mind that the Israelis will check if you have a reservation directly with the hotel. Also, the Egyptians only issue Sinai-only stamps at this crossing free of charge, so if you want to go to Cairo, they'll make you pay $25 for a visa and anywhere from $100 for "visa support"(handwritten note from a "travel agent" without which they won't give you the full visa... You can avoid the whole visa nightmare by getting an Egyptian visa in advance (from an embassy/consulate) - either in Eilat (100 shekels, 2014) or in your home country - prices: in London (GBP20, November 2017), Schengen countries (EUR38, November 2017).
Ferries: There are currently no ferries from Hurghada to Sharm or Ras Mohammed (as of March '17). ABMaritime is planning to open a route there though.. From/To Jordan: There are ferries from Taba and from Nuweiba to Aqaba (Jordan), if you want to get from Sinai direct to Jordan. Their schedules are not reliable and you may have to wait for long hours for your ferry to depart. The ferry from Nuweiba is run by AB Maritime and costs about 70USD for the 3-hour ride.
By plane: There is an international airport in Sharm which has surprising cheap flights to all Europe, Russia and the Middle East. Be aware of the taxi drivers at the airport who always try to cheat you.
There are a lot of checkpoints at each junction where they will ask the driver why he has foreign people with him, so it is hard to get lost. Of course, you hear from time to time bad things of kidnapping etc. in the news but this is mainly in North Sinai. In the South the people are in general all very friendly and helpful. Take always enough water with you, Sinai is a desert. And be aware from this crazy wild dogs everywhere.