When you're moving, it's a good idea to get the sturdiest boxes you can. This helps eliminate the problem of crunching, in which your boxes are crushed by the other boxes stacked on top of them. Sturdier boxes might not be the cheapest, but they'll be worth the expense in the long run.
If your budget for moving supplies is tight, check with a local grocery store for corrugated cardboard boxes. Often, all they do is incinerate them or throw them away, so you may be able to rescue a few and save some money at the same time. Most stores will be happy to help.