First, a few guidelines for working with old sweaters for the following projects:
When cutting a sweater, be careful to keep it from unraveling. Most small-gauge machine-knits, like thin store-bought sweaters, will probably be fine, but for big and chunky, you'll want to use the existing hem or cuff as an edge when possible. If there's no way to avoid a raw edge, you can fold it over and secure it with hot-glue. If you're more craft-oriented, sew a firm line of stitches about half an inch or so from the cut edge. There are some good tips for crafting with old sweaters here, and they don't just apply to sewing projects.
For nearly all of these ideas, it's easy to swap out sewing for hot-glueing when the instructions call for it. And felting — when you shrink something woolen in the hot washer and dryer, the way you've probably actively avoided doing your entire life — will make the fabric thicker and much less likely to fray.
Bonus pro-tip: do not use a sweater somebody handknit for you or they will never speak to you again.