Petra is an archeological site located Outside Israel , In jordan and you can enjoy a day around the area.
If you don't know nothing about petra i will say that this place was the capital of the Nabataeans (you can find also nabataeans places in Israel - Shivta , Ovdat , Mamshit and of the Perfume road.
The word Petra is from the greek work Petrus which means rock.
A trip to petra should cost something like 150 USD for a person.
Try to ask on the hotels or search for a traveling agency.
Texas Ranch is a horse farm where you can rent the horses for a desert ride.
The horses are well train and easy to handle even if you are not an expert.
Use this possibility to have unique experience to have a ride in the desert.
The location is just opposite from the Dolphin reef, before the Underwater Observatory.
A little known place not far from Eilat offer you a great view of a flock of 300-400 flamingos. Instead of going all the way to Africa, these flamingos stay here all year round.
To get here, you need to drive north from Eilat on road number 90. A little after the customs checkpoint, look for a brown sign with the name "Sharsheret Wells". This should be on the right side of the road. Take a right turn onto the dirt road and follow the signs for the flamingo viewpoint. These signs are in Hebrew but you can see a little bird (not a flamingo!) on the top left corner of the sign and an arrow pointing to the viewpont. Eventually you get to some pools separated by dirt dykes. The flamingos are right here. You can get with your car all the way to the pools, no need for a 4x4 car.
Visit the Haibar, 1 hour north of Eilat, to see some animals that are re-introduced to the desert, and some other endangered specious. Next to Kibutz Yotvata.