All our garage stuff is in plastic totes, and we've moved twice with those and never had any problems with movers not wanting to use them and very little damage to the totes themselves. We buy the kind that have the lid split in half that flip over - the sides are structured and they're more sturdy than the kind that have the purely straight sides. We stack those things as high as we can reach with no problems.
When I move, I buy boxes online from Uline. I get a ton of small boxes for my books (I have about 1500 books) and medium & large boxes for the rest of the household stuff. I like having just 3 sizes of boxes to deal with rather than tons of random boxes; it's easier to stack them and move them. I use small boxes for the books so I can still manage them myself after they're filled up.
I also buy a roll of red tape that I use to seal any boxes or mark any totes that I want to move myself - things that are extra fragile/valuable/sentimental. It's really easy for the movers that way - red tape means don't touch!