Cairo is a the capital of Egypt. It is a very large and constantly congested city.
When the Academy of Free Travel established a hitchhiking base in Cairo in Winter 2008/09, they did not hitchhike from the city itself, as it is too sprawling. Equipped with a schedule of cheap trains (for which one must go directly to the train, for ticket booths will not sell to foreigners), they traveled by rail to nearby locales and hitched from there.
It is best to get to the Suez tunnel under the canal, and hitchhike from there. You can take a service taxi to Suez from the depot next to Ramsis train station in cairo (8 EGP), and ask to be dropped off in the junction where you can turn to Sinai, around 10km before Suez. All the traffic headed to Sinai from Cairo has to pass here.
You can also go to the small train station next to the Ain Shams metro station in Cairo, take a morning train to Suez. One can try to hitch east from there. The Cairo-Suez train runs every three hours and costs about 5 Egyptian pounds.
North to Alexandria
In the city there is a big bus station called Abd el Moneim Riad which actually has two separate areas. Make your way to the second one which is closer to the river and find a bus going to Lebnan Square. Here you can catch basically any minibus and ask to get out at the Cairo Alexandria Desert Highway. There is an actual bus stop here and a bunch of rugged stone steps that will take you down to the highway. From here you can start hitchhiking or choose to walk about 5 kilometers to the toll plaza. If you do hitch, any vehicle is good as a little over 4 kilometers up the road is a massive shopping mall called Dandy Mall. And from here it is just a short walk to the toll plaza which can be seen from the mall.
There is probably also a way of using public transit to reach the shopping mall if you found the right person to ask.
ANother option is to find a bus going to Faisal which is quite close to the entrance for the pyramids. Ask around for Remaya Square as this is basically where the Cairo Alexandria highway starts. You'll probably want to walk a bit further up the road though as there is still a lot of local traffic and it can be pretty chaotic. But same thing as Option 1 in that eventually you can reach Dandy Mall and the toll plaza where you should be able to find a direct ride to Alexandria if your patient.
South to Minya, Asyut, Sohag, Luxor
It is best to really get out of the city to start hitchhiking, as most people don't go far. A good option is to take the metro to Helwan (last station, 1 EGP) and walk South on Mansour, through the market. You will pass a gas station (Misr) and a big roundabout. After 800 meters you can take any microbus going further South. It should cost you 1 EGP. You could also walk the remaining four km's to the highway. Get off at the intersection with the Autostrad. From there you can start hitchhiking. There will be more people waiting for shared transport. Ignore them and walk a bit further. Aim for the better looking cars, others might ask for money! Always make sure you don't have to pay, beforehand!
Most cars will be taking the pretty new highway called El-Karimat El-Zafrana. It is being extended until Luxor. This road is in a good condition and people will stop for you on the way. Try not to leave the highway if somebody drops you and you want to go further South.