There's a lot of great exercises you can do with a jump rope. What I love about the jump rope is it's a great cardiovascular workout and you can pretty much take it anywhere. If you travel a lot, throw it in your suitcase. Throw it in your carry-on bag and you've literally got a gym with you.
The things you want to think about when it comes to jumping rope is the length of your jump rope. You can test it out by standing with your feet hip width apart and just make sure it comes to about chest height. If it's too long for you, then you can just wrap it around, tie it in a knot. You want to make sure that it's just level.
And then, talking about basic jumping mechanics - rolling through the feet, toe, ball, heel. From the side, just keeping in mind that most of the action here is not from the entire arm, but it's really just at the wrist. You're still going to sculpt your arms here because you have to do a lot to stabilize your shoulder joint. If you wanted to use a heavy weighted rope, that's when you would be getting a little bit more of the upper body work. If you're using a light, like a speed jump rope, it's really just for cardio purposes.
A basic jump, you could even do a double jump in between if you needed to slow it down and work on getting the coordination. Then once you get good you start picking up the pace. You could even do a single leg hop, and then switch to the other side. You could do alternating, like jogging in place. You could do an in and out jumping jack motion with your legs. You could do a cross jack. There's just a lot of different ways to add variety, add coordination.
Once you get really good, you can start picking up that speed and skipping rope. You could do a double swing where the rope is going around twice and you're only jumping once. Crossing. Again, really just having fun with it and keeping your heart rate up.