Climbing in Sinai
The main climbing area in the Sinai desert (in Egypt) is around the town of St. Catherine's. There are hundreds if not thousands of trad routes waiting to be opened on beautiful red granite rock. There is a guidebook online, available for free, but it is not updated often, and might have mistakes. The climate is desert-like with the added factor of the high altitude (1500m) making the weather cooler and even bringing some snow in winter. The best seasons to climb in this area are Spring and Autumn (Winter is cold and Summer is hot).
The monastery of St. Catherine's is nearby. It draws many Christian tourists, most of which head up Jabel Mussa (Jabel means mountain in Arabic) to see the sunrise or sunset.
To reach Egypt you can fly to Cairo or possibly to Sharm El Sheikh, or go by land from Israel. St. Catherine's is located about 3 hours by car from the border with Israel. Also, by passing through Israel and continuing directly to Jordan (20 minutes away from the Israeli-Egyptian border), you could visit the spectacular Wadi Rum sandstone climbing area (about an hour from the Israeli-Jordanian border).
The rock is mostly Granite, but there is also some Gneiss. The routes are mostly cracks and slabs. The cracks range from micro finger jams to wide chimneys. Rely on traditional protection, there are very few bolts. Friends, hexes and tri-cams are very useful, nuts are not as useful. Always take extra slings or rope with you to rappel. Long slings for protection could be useful, as would a double rope.
Sleeping & Eating
There are a few cheap hostels/campgrounds in town (see the guidebook). They also serve meals by request. There are some shops in town where you can get basic supplies.
This is prime hiking territory too. For most trips the regulations require you to take a guide ("dalil" in arabic). You can try and get away with not taking one, but you could get in serious trouble.
Sinai's beaches are beautiful. Lazing on the beach is a popular pastime for young Israelis who flock down to the beaches on holidays and weekends. Most of the accommodation is in very basic cane bungalows right on the beach, but hotel rooms are also available and cheap. The snorkelling is astounding and scuba divers claim the diving at Ras Muhammad Marine Reserve rivals the Great Barrier Reef. Cairo and the rest of Egypt are not far, where you could visit the pyramids and explore further.