Do I have to resample images when I'm resizing them? I want to make them smaller.
Yes. Changing the size of the image without resampling is called "cropping" — the pixels in the new image are identical to the pixels in the old image, they just live in a differently-sized space. You want to map the old pixels into the new space, and that requires resampling whether the new space is larger or smaller than the old space. The reason why resampling is optional in the resize dialog of some editors is because you may want to change the pixel resolution of the image — not the number of pixels in the image, but the number of pixels per inch/cm. In that case, you're not changing the image data itself, just the image metadata (the information about the image).
You don't strictly have to re-sample if you use an exact multiple (you could simply drop every other pixel) but it is silly not to re-sample as it results in a loss of overall quality and the production of artifacts if you don't.
When you shrink an image, even at an exactly even multiple, the best image quality is maintained by combining the information from the pixels being removed in to the pixels being kept (down-sampling). If one pixel was black and the other white, then the pixel that ends up existing where they both were would have been grey, not black or white. Putting a black or white pixel in the place would result in an artifact that doesn't match up with what the original image looked like.