Once you have a visa, it is a pretty ride. The ride from Jerusalem on Route 1 down to the Dead Sea should take about 30 to 45 minutes. Have your camera ready for crossing Sea Level, and on the left you will see a large marker when you reach 300 meters below sea level.
After you turn left on Route 90 for the short ride north to the Allenby Bridge, if you like geology, there will be great hoodoos on your right. Not always, but it would not be unusual to see a herd of camels on the left as well. The terminal between Israel and Jordan is at the foot of the bridge, and you should easily find transportation on to Jordan.
I'd bring water.