Just think about it. What is it called when you put ink on your finger and then press that finger onto a piece of paper? A fingerprint. Or what about when you step into wet cement? That's a footprint. When you put your dirty hands on your clean shirt? Yep, that's a handprint. Printing is as easy as that!
First things first. This will be messy. Wear old clothes that are meant for getty dirty. Then, spread out some newspapers or a plastic sheet. Make sure your mess stays in that area!
Hand printing and finger painting requires no tools--just paint and hands. Pour a little paint onto a paper plate to cover the palm of your hand, or dip your fingertips into a small bowl. On a piece of blank paper, press the paint from your hand to the paper. Try patting, squishing, dragging, or swirling for a different look.
If you're comfortable with finger painting, try making a stamp out of a potato! Scrub a raw potato clean, cut it in half, then draw a shape or a letter onto the cut side. If you're old enough, use a knife to cut away the potato from the outside of the lines to the edge, leaving a raised surface in the middle. An adult should always help when knives are involved. You could use a cookie cutter, too--just stick it in the potato and cut around the edges! When you like the shape, dip it in a little bit of paint, and stamp all over a blank piece of paper. You can make pictures, cards, or even wrapping paper this way.
If you really really like printing, then you should learn about a man named Johann Gutenberg, who can claim most of the credit for inventing the tools to print books quickly and inexpensively. You know what's amazing? He invented those tools almost 600 years ago, and people still use them today!
Koscielniak, Bruce (2003). Johann Gutenberg and the Amazing Printing Press (Gr. 2-5)
This is a history of books and history of printing, all in one. From the earliest days when making a book meant writing each word by hand--all the way to the first book ever printed from moveable type.