Once in Egypt, getting around is easy and relativly cheap on Egyptian Airlines, however, finding a flight from Tel Aviv to Cairo will be difficult (Arabs and Isreal don't mix well) and there appears to be no direct flights. Also, there is something about getting the correct visitor's visa went crossing via Eilat, if you get the wrong one, you cannot go from the Sinai to mainland Egypt (west of the Suez canal).
EgyptAir
You can fly or bus down to Eilat which is resort/border town with Egypt, walk across the border and catch another bus down to Sharm El Sheik (4 hours) then fly to Cairo for about $75.00 USD.
Do some research about Dahab, which is on the road to Sharm. It is a nice inexpenisive place and with decent diving, plus they have a blue hole to check out.
dahab hotels, safari and diving - Dahab Red Sea
Have fun.