Is my cassette or chain worn out?
Use a chain wear gauge to see if the chain needs replacement (e.g. Park Tool CC-3.2).Generally, these err on the side of replacing a bit early, but if you want to be pedantic about it, you can use the Sheldon Brown link below to measure with a ruler. 

As for the sprockets or chainrings, I'll defer to this page by Sheldon Brown. The upshot is that worn teeth look like ramps and a non-stretched wont sit properly in the sprockets as shown in the image below (from Sheldon Brown)