Israel's only Red Sea coast is at Eilat. There are several beaches in the town where big hotels have put out their sunbeds and parasols. If you are not a client of those hotels you will have to find a little corner of sand to lay down your towel. If you go to Eilat by car it's best to get away from the centre of town and take the road which leads to the Egyptian frontier. Along this road there are private and public beaches.
If you like diving, the closer you get to the Egyptian border, the more beautiful the sea beds are. If you like walking, I recommend you spend the day in Timna national park which is 25 kilometres to the north of the town. There are walks surrounded by a countryside made up of red rocks and red mountains.
If you are planning on a trip to Israel that lasts more than a week then you can easily spend 2 days on the shores of the Red Sea.