JordanYou need your passport and plenty of money, because it is equally expensive as Israel. There are three border crossings, near Bet Shan in the North, near Jericho and in Eilat. You can obtain your visa at the border in the case of the first and the last crossing. Crossing the border is problem free, but on the way back you may be carefully searched.
EgyptThere is only one border crossing to Egypt. It is located in Eilat. You do not need visa if you plan to visit only Sinai peninsula. If you plan to visit rest of Egypt then either you can arrange visa in Egyptian embassy in your home country, or in Egyptian embassy in Tel-Aviv (54 Basel St. (+9723) 5441615) or in Egyptian consulate in Eilat (68 HaAfroni St., (+9728) 6376882), or, at least theoretically, in Issta travel agency on campus. As in the case of Jordan, crossing the border is very easy, but on the way back reserve extra time for a potential search. On the Egyptian side there is a small touristic village Taba, where you can change money and catch a bus or taxi to mainland Egypt or to a Red Sea resort.
Egypt still is very cheap. Be ready to pay from 5 to 10 USD for a hotel room (no air-con) and from 5 to 15 USD for food per day. Public transportation is cheap but not always working. The photos below were taken in Dahab resort on Red Sea, some 150km south from Eilat. Sinai peninsula is a reasonable target for a 4 days or longer trip from Beer-Sheva.