The packing boxes you get for your move should be able to handle a load of at least fifty pounds. That's the maximum recommended limit for any box, for safety reasons. If you try to save money with flimsier boxes, you're likely to regret it later.
When packing, avoid wrapping your things in old newspaper. The ink can rub off and stain your belongings. Newspaper does make a great packing material, but it's best if you purchase blank newsprint. Most moving companies can provide some cheaply.