The Three Ball Cascade is the most
basic juggling pattern, and the first trick any would-be juggler should
learn. The Cascade is generally considered to be the easiest pattern, and
forms the backbone of many other tricks.
To begin learning the pattern, it is important to make sure you are comfortable with throwing and catching a ball. It sounds silly, I know, but when first starting to juggle I found that my left hand (non-dominant) had a hard time accurately throwing the ball to my right. You will want to get very comfortable throwing the ball from you dominant hand to your non-dominant, and vice versa. Try to keep the height of the throws a little bit above your head. Once you have mastered this, you are ready to move on to the next step.
It is at this second step that most jugglers begin dropping (short hand way to say that you have failed to catch your ball) a lot. As such, this is probably a good time to remind you that using tennis balls on a hard floor is not the best way to juggle. For beginners, I would highly recommend getting some good quality bean bags such as these ones from Dube. They are soft, don't bounce or roll (much), and are quite easy to catch.
Now that you have the proper juggling equipment, it is time to add in the second ball. This second ball is going to travel through the air with a similar trajectory as the first ball, except it is going to be thrown underneath that first ball, and travel in a different direction.
To start, hold a ball in each hand, and then throw one ball up in the same trajectory as step one. As that ball reaches it peak, you are going to throw the second ball up underneath it towards your other hand (the one that threw the first ball), then catch them both. You are going to want to make sure that you practice throwing the first ball from both hands. If that description confuses you, just focus on the animation above and try to copy its throws. This pattern is known as the Two Ball Cascade.
You must make sure not to pass the second ball horizontally to your other hand (shown below), as this is the way people often intuitively try and juggle two balls.
Once you are able to juggle two balls correctly without much trouble, you are ready to add in the third ball. The first pattern to learn with three balls is the flash, where every ball is thrown up and then caught again like so.
The flash is very similar to the Two Ball Cascade, except you sneak in a third toss during the pause that occurs after you throw the first two balls. While the timing of this trick is slightly different than the standard Three Ball Cascade (notice there is still a period of time where all three balls are in the air), practicing the flash will improve your accuracy and throw speed. Once you can do the Three Ball Flash reliably without too many drops, you are ready to move onto the final step.
For the final step in learning the Three Ball Cascade, you will be combining all the skills learned thus far. Starting with three balls (two in your dominant hand), you are going two throw your first ball, and wait for it to reach its peak. Then you will throw the ball in your non-dominant hand, allowing for the first ball to land in your now empty hand. Once that second ball reaches its peak, you will throw the third ball from your dominant hand, clearing room for the second ball to be caught. And so the cycle continues, each time a ball reaches its peak, another one is thrown, freeing up a hand to catch the first ball. The animation below shows the pattern being run for five throws and catches.
Now that you have the basic pattern down, the only thing you have left to do is practice, practice, practice! When it comes down to it, that's all juggling is. No one is born good at juggling, the experts work hard for the talent they posses, and anybody can become as good with enough dedication and work. After practicing the Three Ball Cascade, you should see your throws becoming more accurate, your arm movements less noticeable, and the height required for you to run the pattern will decrease.
Congratulations, you are now an official juggler. But this is just the beginning, there are still countless more patterns out there for you to try. Browse through the Library of Juggling to find some cool new tricks to learn. For a beginner, I would recommend Juggler's Tennis and the Half-Shower next. Good luck!