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. I used regular Hefty bags for a two-block move many years ago, and all my towels and washclothes wound up in the trash by mistake. I think you'll find that large, clear trash bags are useful to have on hand even if you don't plan to use them for packing. When a box is packed, move it to the staging area, so you won't be tripping over it later. Heavy boxes go on the bottom, lighter boxes and breakables on top.