By plane
Eilat Airport ETH is right in the middle of the city. Flights to Tel Aviv are frequent and take only 50 minutes, but expect to pay around NIS 250 for a one-way trip. However, tourists can arrive in Eilat on charter flights via the Ovda International Airport VDA, also known as Uvda, 65 km 40 miles and nearly a 50 min drive from town.
A cheaper way to get from Europe directly to Eilat is via the nearby Taba International Airport in Egypt. Charter flights to Taba are operated by several airlines, e.g. the German "Condor" on Wednesdays from/to all major German airports. Taxis from Taba airport to the Israeli border station at the Hilton hotel run for max. 150 Egyptian pounds 40 min, from there a taxi to Eilat is around 25-30 NIS 10 min. Or take local bus 15 6 NIS. Border crossing normally takes less than 30 min. On arrival at the airport insist on Egyptian "Sinai only" visa, otherwise you're charged 15 USD visa tax. Note that Egypt charges 75 LE tax when leaving Taba coming from Eilat.
By car
There are a couple of ways to drive from Tel Aviv to Eilat. One is via Mitzpe Ramon. Another nice alternative is from Tel Aviv to the Dead Sea via Arad, stay a couple of days there or just make a short stop and then continue to Eilat. It takes approx 5 hours from Tel Aviv.