Because almost all the residents of Taba are involved in tourism, all transportation passing through Taba is meant to take tourists somewhere for a price. Right after crossing the border everyone will tell you that there is no traffic, no buses and no shared taxis - only overpriced taxis. Just ignore these people and continue walking.
Hitchhiking out for free is difficult, in fact nearly impossible. One can save money, however, by avoiding the Taba entry tax (120 Egyptian pounds as of November 2017). Instead of walking out of the city through its only road, turn down the alley next to the bus station and walk down to the beach (you can pass through the gate there). Proceed along the beach for a kilometer until you are out of view of the entry tax station on the main road. You may then be able to get a lift 9 kilometers up to the police roadblock where the road splits west towards Suez and Cairo or south towards Nuweiba, Dahab, and Sharm el Sheik. This checkpoint is probably the biggest hassle on the way out of Taba - the police will hassle anyone who isn't a taxi but takes a foreigner and also a pretty nasty one, since the Suez road leads to North Sinai and the Police don't act nicely and interrogate even the natives why they are going that way, so the general atmosphere can be a little tense. From the roadblock you can wait for a shared taxi to arrive, which will likely charge a somewhat lower price than the bus company in Taba. The police will not tolerate your presence there if you are seen turning down taxis, so do not insist on getting a lift for free. A bus to Dahab should cost around LE50. To avoid hassle with the police (both for you and your driver), it is recommended you take a taxi/bus at least to Taba Heights/beyond the main Taba/Suez/Nuwaiba checkpoint.
uncle_sam01: arrived to the main Taba-Suez-Nuwaiba checkpoint in two nice Bedouin guys' car from Dahab (which the police in Dahab actually hitched for him) yet the police at the checkpoint in Taba were quite agitated (yelled at the Bedouins for taking a foreigner without a taxi license) and did not allow him to walk towards the border crossing or hitch, saying a bus will come. They kept him there for 2 hours and eventually a passing police truck took him to the border. November 2017