When choosing a moving company, you want to know which moving services you might want to pay more for and which costs you can cut. A very cheap mover may well just be some big guy and his cousin who will rent a van and come over and move you. If you're a student, or if you don't have much of value, this may actually be a great option for you; but if you have any special needs, delicate furniture, or breakable items, you definitely do not want to go there.
Any good movers will list all of the equipment they have expertise moving, all of their certifications and qualifications, as well as all of the equipment they own. You may not need most of it, but you'll be paying more for the fact that they have it, anyway. You want to find a mover who suits your needs but only charges you for the moving services they're actually providing for you.
There are some relatively large moving companies that specialize in cheap moving. One of the best ways to find them is by finding a moving company that will itemize their costs instead of citing you a lump sum. The ideal company will charge you for costs, and then a small margin of profit that you can assess by the itemization on your bill.
If a mover quotes you a price before finding out exactly what items you have to move or talking to you about strategies, they are probably not the right company for you. As with any service industry, movers should understand any unique details of your situation, and then tailor a custom solution. Remember, ask and you shall receive. Say something like, "That figure is a little bit out of my price range. Is there anything we can do to bring it down?" You may be pleasantly surprised by the answer you get.