Israel has one international airport: Ben Gurion in Tel Aviv (TLV). Depending on when you want to travel, you may find good deals to travel abroad. Students have traveled to Jordan, Egypt, Turkey, Cypress, Greece, Italy, Spain, France, Switzerland, Romania, Hungary, Austria, Germany, Holland, Denmark, Poland, Georgia, India, and Thailand, among many other places.

Great flight search aggregators:
Google flights
Low cost airlines:
Wizz air has a hub in Budapest, which is nice to visit, and its also a good jumping off point to the rest of Europe
Easy jet
Ryan air
Last minute travel deals:
Daka90 (Hebrew- translate)
Issta (Hebrew- translate)
Trains within Europe
Seat61 is an excellent aggreagator of trains throughout europe as well as ferries

Jordan (Aqaba, Petra, Wadi Rum, Amman, etc.) - Take bus No. 390 or 394 to Eilat (4 hours, ~50NIS). *When you are nearing Eilat, ask the bus driver to let you off at the Yitzhak Rabin crossing. You’ll then walk about 10 minutes to the Jordanian border. If you miss this crossing, or first want to head to Eilat, take the bus to the Eilat city center, then a taxi to the Jordanian border (10 mins, ~40NIS). Taxis cost ~8 Jordanian Dinar to Aqaba Center. Be wary of dishonest Jordanian taxi drivers trying to drum up business, claiming there aren’t busses to Petra/Amman.

Egypt (Cairo, Dahab, Taba, Egyptian beach camps, etc.) - Take bus No. 390 or 394 to Eilat (4 hours) and a taxi to the Egyptian border (15 minutes). There are public buses, mini-buses, and taxis once you cross the border. It's another 3-4 hours to Dahab by public bus. Dahab is a great place for diving. It's an 8 hour bus or mini-bus ride to Cairo from the Israel-Egypt border. The approximate cost of all roundtrip transportation to Dahab city (including buses in Israel, taxis to/from the border, border crossing fees for each country, shared taxi to/from Dahab) is around $100-150 USD. Note that you will need to get a visa to go anywhere in Egypt besides Dahab and the Sinai. Visas can be acquired at the embassy in Tel Aviv or at the consulate in Eilat.

Pro tip: It's a good idea to check your embassy website for travel warnings