Movers charge by the hour. A variety of boxes of different shapes and sizes will slow down loading the truck. What you save in boxes you will probably lose in movers’ costs because your assorted boxes will slow down your movers.
You want boxes that are stackable, and uniform in size and appropriate to the items you are packing. You will need large boxes for blankets, pillows, linens, and large items. Books and heavy items need to be in smaller boxes. Boxes that are too small will increase the number of boxes the movers have to handle, and will add time and therefore increase the cost of movers.
Boxes will be carried to the moving van and stacked on top of each other. The boxes that hold computer paper are notorious in the moving world for collapsing when stacked. Avoid those boxes. If you press down on a fully packed box and the box gives in, you haven’t filled it tightly enough. Your boxes need to be packed full and taped shut.
Use lots of paper when packing breakables. Ask your mover for a brochure on packing properly. Newsprint will rub off on your hands, your linens, and your dishes. Save yourself the mess and extra work by purchasing unprinted newsprint.