Suez is a city in Egypt.
East to Sinai, Israel
From right next to the train station at the outskirts of town, a road goes north several kilometers to the tunnel running under the Suez Canal. Cars will not stop for hitchhikers on the road just before the tunnel, believing it is forbidden to take foreigners across. However, if you walk to the police roadblock in front of the tunnel entrance, they may be willing to put you in a vehicle bound for your destination in Sinai.
September 2018: TheLoneBaker tried to enter Sinai by doing the above and walking to the toll and was told she had to go back and find a vehicle to ride in before the checkpoint. Ended up standing just after the exit of the military zone/police checkpoint which was out of view of the toll itself. However, when brought to the toll, they checked passports and then forced TheLoneBaker to exit the vehicle and insisted she had to take an expensive bus or not go at all.
Might be better to find a bus at the main intersection before the tunnel and then ask to pay a couple pounds to ride the 10 kilometers through the tunnel and to the main highway on the other side.
West to Cairo
Instead of hitchhiking, it's much easier to take a train to Cairo, which costs only 5 Egyptian pounds.
Update September 2018: According to a local, there is only one train a day going rob Cairo and it leaves at around 6 in the morning. It costs between 7 and 10 pounds. (They weren’t quite sure the actual price.)