Banana boxes often have juice from leaky bananas on them. They are very strong but they are not my first choice. I prefer apple boxes. Tell the grocery store ahead of time that you want a bunch of them with lids, and to please save all the cushion liners. Those liners are amazing for cushioning the bottom of a box or using between breakables.
Once I wrapped a 2000 year old antique vase in a heavy layer of newsprint and then wrapped it again with those cushions from the apple boxes. Are use lots of tape also. Then I placed it in a big box with more cushioning, just in case. I don’t think that verse would’ve broken even if we had dropped the box on concrete!
If you will be storing any of your things after the move, be sure to use storage tape rather than packing tape. There’s a huge difference in quality, but the price difference isn’t a dealbreaker.
Liquor store boxes can be great for packing glasses. Be sure to get the separators that came with the box.
One of the most expensive parts about moving is buying packing paper. Somethings simply will not travel well in anything else, and it’s very difficult to purchase packing paper of a size large enough to actually make it useful. The kind U-Haul sells is really small-and expensive! If you do your work ahead of time you may be able to find large sheets of packing paper for a better price. Bubblewrap is even more expensive, but sometimes you can find it used (and free) in not too bad of shape.
I worked for a natural food store for a while and I was astonished at all the lovely packing material that they received and had to throw away. (The stuff I saw was heavy paper with a crinkle pattern punched in it in to make it soft and thick at the same time.) You might be able to find packing material for free if you ask around.
Sometimes people think they can save money using used packing paper. If it doesn’t have many wrinkles in it, and you use more than you think you need, you may turn out all right, but you take a chance of having broken stuff.
Many places such as U-Haul offer a booklet on how to safely pack different items. Get one and study it.