If you want to see and do, you have nothing to look for in Eilat. Eilat is a resort city where you visit the pool, the beach and the restaurants.
I suggest staying 2-3 nights in Jerusalem and the rest in Tel Aviv - including taking some day tours to the North (Haifa, Nazareth, Sea of Galilee). If you want to travel on your own, a night at a northern hotel is a great option.