There are many things you can use to carve stamps - one is called Easy Cut and it IS easy to cut. The downside is that it crumbles easily. I did this stamp (on right) years ago and the corner's have broken off, so don't know if I would use this material again.
Another material is linoleum (below, left) and quite difficult to cut. There is also "Easy to Cut Lino" which I haven't tried yet but am sure is easier than regular linoleum.
A great, fun and easy to cut surface is a gum eraser. The pink erasers are more dense than gum erasers and probably hold up longer. The main drawback is the size - very small so they have their limitations.
"Mastercarve" is my favorite material for carving. It's available in art stores and often craft stores. It's easy to carve and doesn't crumble. You can buy a big piece - 9" x 12" or smaller pieces. Another plus - it's thick enough so you can carve every side, which we did making it more economical.
You can also use rubber gaskit material from the plumbing department of a home improvement store. It was an interesting material, a little more difficult to cut and quite thin so you could only use the one side, but reasonable in price.
We all used the Speedball cutter which is easy to use and find. I like the fact that it has many interchangeable tips for cutting different width lines.
Designs for stamps was next - wow, the ideas are endless. If you can draw, you can rely on your own abilities. You can also use books or other sources - they're everywhere. One idea was to scan a photo, put it in a photo editing program and turn it into a line drawing. The Dover coloring books full of copyrite free images is another possibility. Below - a page from one of their books.
There were several ways to transfer the image for carving. One method was to use an image printed from a laser printer. You need the toner so an inkjet printer won't work.
Place the image face down on block. Wet a cotton ball with Acetone - nail polish remover and gently rub over the back of the image.
The Acetone will release the image onto the block - not perfect but clear enough for carving!
There are times when using a laser image isn't convenient and you want to do it NOW! An alternative to the above is to color the back of an image with graphite. Everyone has a pencil around the house, so this is easy. If you have one with exceptionally soft lead - even better!
Turn the image over, place on material to be carved and using a stylus type of instrument, firmly press along the lines of the image.
The image is transferred very easily.
One of my favorite things about hand carved stamps is something called "Eraser Trails". When carving away the negative space to make the image protrude more, oftentimes, there are lines left behind. It's impossible to get the negative space perfect. These lines show up when you print the stamp.The stamp below is one I carved awhile back with eraser trails on it.
To stamp, put paint in a recycled styrofoam plate or on a piece of freezer paper. Roll a brayer into the paint to cover it but not too thick; apply the paint to the stamp...
and press onto fabric. Sometimes, it's more effective to lay the stamp down as you see above (right side up) with the fabric on top and press. Both produce an image but the latter seems to have a darker image.
Another hint is to use a purchased stamp and....
apply paint. Next, press the stamp onto carving material
and carve a stamp using this image. I know - why would you do this with a pre-cut stamp. Well, if you love the look of the "Eraser Trails" and happen to like the pre-cut image, it's an interesting way to go. I do love the look of handmade stamps. Purchased ones are nice, convenient and have their place, but there's nothing that can replace one made with your own hands.