I always used to use newspapers to wrap dishes and other breakables. Newspapers are plentiful and the price is right, but I always wound up washing every dish I owned, because they'd all be smudged with ink. Now I use plain white tissue paper, the kind you'd find in a shirt box or a gift bag. You can generally find it in the gift wrap section for as low as 20 sheets for a dollar. Whenever possible, put heavy things in small boxes.
Big boxes are fine for pillows, baskets and blankets.
With a little care, you can safely pack your smaller framed pictures and artwork among the pillows and blankets, by the way. Instead, label your boxes to within an eyelash of destiny. All this labeling may seem like overkill, but it really speeds up moving in and unpacking. Some people use trash bags for bedding, clothing and whatnot. If you plan to do this, I strongly recommend that you use the large clear leaf bags so you can see the bags' contents.