I am assuming that by bedding you mean a duvet cover and/or comforter. I do mine by hand in the tub and hang outside to dry. Same way I always do my sleeping bags. Use a gentle soap like Woolite or Ivory Snow so it rinses easily. For down use a soap made specifically for down if you can.
mattress pad is the bigger issue, I don't use the comforter, and the duvet for the dog is just a light blanket.
It's amazing to think my first fiancé and I never had bigger than a full, and were never uncomfortable. Second significant other is when I went to queen, and was more than enough. I didn't get a king until it was just the dog and I, he's been the biggest bed hog of all, always has to be right in the middle and I know he understands the laying lengthwise thing, but he's a stubborn fuck