A must-see, the Dead Sea is beautiful and fun, but be careful when swimming. At 30% salinity, nearly 10 times that of the normal ocean, the Dead Sea is the saltiest place on Earth. The water level is also dropping a meter every year-- meaning the water will only get saltier.
The easiest way to get to the Dead Sea from Amman is by taxi. It is a little less than an hour and a half to get to the Dead Sea, and is affordable when split among 3 or 4 people.
Otherwise, bus is also an option. Check bus schedules carefully, however, as the Dead Sea is not terribly crowded, and bus runs aren't frequent.
Swimming in the Dead Sea
Before you go:
Try to avoid shaving for at least three days before entering the Dead Sea, and try to avoid having any open wounds (blisters, cuts, etc.) Make sure to take plenty of water with you. If you have water shoes, you may want them-- the salt on the bottom of the lake can be sharp. Flip flops are not recommended, as they will get stuck in the mud.
Entering the Dead Sea:
Walk in slowly, careful not to splash. When you get to about waist deep, simply sit back, and you will easily be able to float on top of the water. Be extremely careful not to get the water in your eyes, ears, nose, or mouth. If you do, have a friend lead you to the fresh water showers commonplace at public beaches. Do not put your head underwater; floating on your back is preferable in order to avoid this. Do not stay in the water for more than thirty minutes, and afterward, wash all the salt off in freshwater showers or your skin will be affected. Be sure to hydrate and flush the salt from your mouth.
You can take a mud bath in the world-famous Dead Sea mud, which contains four times as many minerals as normal ocean sand. You can feel the mud on the sea floor. The extremely slimy stuff is the best. It's safe to smear it all over your body, and don't forget to snap a pic! Be warned that the mud may stain light colored swimsuits, and if you are fair skinned, it may tint your skin orange.