The Book of Mormon is quite clear as to the course they took before building the ship. First they traveled almost due south from Jerusalem to the valley of Lemuel by the borders of the Red Sea (probably near the Gulf of Aqaba). Next, they traveled in a south-southeast direction.
Hugh Nibley
"As to the direction taken by Lehi's party the Book of Mormon is clear and specific. He took what we now know to have been the only possible way out, what with immediate danger threatening from the north, and the eastern and western lands held by opposing powers on the verge of war. Only the south desert, the one land where Israel's traders and merchants had felt at home through the centuries, remained open--even after Jerusalem fell this was so. And the one route into that desert was the great trade-road down the burning trough of the Arabah. For a long time the party traveled south-southeast and then struck out almost due east over a particularly terrible desert and reached the sea at a point to be considered later. Nephi is careful to keep us informed of the main bearing of every stage of the journey, and never once does he mention a westerly or a northerly trend. The party traveled for eight years in but two main directions, without retracing their steps or doubling back, and many of their marches were long forced marches."
"All this entirely excludes the Sinaitic Peninsula as the scene of their wanderings, and fits perfectly with a journey through the Arabian Peninsula. The slowest possible march "in a south-southeasterly direction" in Sinai would reach the sea and have to turn north within ten days; yet Lehi's people traveled "for many days," nay, months, in a south-southeasterly direction, keeping near the coast of the Red Sea all the while. Ten days take a foot traveler the entire length of that coast of Sinai which runs in a south-southeasterly direction--and what of the rest of the eight years?" (Lehi in the Desert and The World of the Jaredites, pp. 54-5)