No Egyptian visa is required, as special 14-day Sinai permits are granted on arrival at the Taba border, Taba Airport and Sharm el-Sheikh's airport. Note that this permit allows travel only on the eastern Sinai coast and the Mount Sinai with St. Catherine's Monastery.
By train
There are trains from Cairo to Port Said at the northwestern edge of the Sinai, but no further.
By ship
Ferries run regularly between Aqaba in Jordan and Nuweiba, bypassing Israel and the sometimes complicated border arrangements. Generally there is no visa fee for entering Jordan through Aqaba since it is a part of the free trade zone. The line to Nuweiba is operated by ABMaritime, see their website for the 'official' timetable and current prices.
There is more information about the boat crossing in the itinerary Ferries in The Red Sea.
By road
There is a busy border crossing between Eilat, Israel and Taba. See the Taba article for details on crossing in either direction.