Origami keeps you in the moment, rewards patience and diligence, and can be done for exceptionally little money.
But unlike all those other fads, origami is something that has stuck with me through life. I have several forms memorized, and can follow any pattern, given a few trial runs to get a feel for it. When I broke my wrist a few years ago and was unable to knit, I fell back on origami to keep my hands occupied while I watched television.
Origami follows that classic rule of being something that is "easy to learn, but impossible to master." The best go on to create their own amazing patterns, or to execute ages-old patterns with an incredible perfection.
Origami keeps you in the moment, rewards patience and diligence, and can be done for exceptionally little money. You can start folding with sheets of paper from the office recycling bin, if you like. The fancy pretty origami paper squares are nice, and convenient, and easy to fold, but they are by no means necessary.
To begin, find a book or set of instructions for the simplest forms. If it shows you how to fold a frog and a crane, you are in the right place! Start by learning and memorizing the graphic symbols which are used on origami instructions. Once you can spot the difference between "valley fold" and "mountain fold," you are well on your way to success. Most books are organized to progress from the simplest forms to the most complex, so work your way through from beginning to end.
As with so many things, your local library is the best place to start. Libraries have a wealth of beginner's origami books. Most books will either have the same stuff as all the others, or be utterly impenetrable, so there's no sense wasting money buying books to start with. If you find a favorite book, buy a physical copy to keep on your shelf.
The best skill to learn at first is how to make a square out of a rectangle of paper. Master this skill, and you will find that the world is full of interesting paper that begs to be turned into a crane, lotus flower, or cat!