@SarahGen-- Are you bending your knees while shaving them? For parts of the body that have excess skin like the knees and the armpits, you need to make sure that the surface of the skin is completely flat before shaving. Otherwise, you're going to get a lot of cuts.
For your knees, bend them and use short, gentle strokes with lots of shaving gel. For your armpits, extend your arm up and make sure that underarm skin is totally flat before using a razor. It takes some practice, so keep trying.
When you do get cuts make sure to wash with soapy water and apply antibiotic cream on the cuts. And always change razors after a cut.